Theorem psubim-P6-L1 789

Description: Lemma for psubim-P6 791.

Assertion

Ref	Expression
psubim-P6-L1	⊢ ([𝑡 / 𝑥](𝜑 → 𝜓) → ([𝑡 / 𝑥]𝜑 → [𝑡 / 𝑥]𝜓))

Proof of Theorem psubim-P6-L1

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		axL2-P3.22.CL 256	. . . . . . 7 ⊢ ((𝑥 = 𝑦 → (𝜑 → 𝜓)) → ((𝑥 = 𝑦 → 𝜑) → (𝑥 = 𝑦 → 𝜓)))
2	1	dalloverim-P5.RC.GEN 593	. . . . . 6 ⊢ (∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓)) → (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜓)))
3	2	imsubr-P3.28b.RC 270	. . . . 5 ⊢ ((𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓))) → (𝑦 = 𝑡 → (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜓))))
4	3	axL2-P3.22 254	. . . 4 ⊢ ((𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓))) → ((𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) → (𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜓))))
5	4	dalloverim-P5.RC.GEN 593	. . 3 ⊢ (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓))) → (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) → ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜓))))
6		df-psub-D6.2 716	. . . 4 ⊢ ([𝑡 / 𝑥](𝜑 → 𝜓) ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓))))
7	6	bisym-P3.33b.RC 299	. . 3 ⊢ (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓))) ↔ [𝑡 / 𝑥](𝜑 → 𝜓))
8	5, 7	subiml2-P4.RC 541	. 2 ⊢ ([𝑡 / 𝑥](𝜑 → 𝜓) → (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) → ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜓))))
9		df-psub-D6.2 716	. . . 4 ⊢ ([𝑡 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
10		df-psub-D6.2 716	. . . 4 ⊢ ([𝑡 / 𝑥]𝜓 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜓)))
11	9, 10	subimd-P3.40c.RC 330	. . 3 ⊢ (([𝑡 / 𝑥]𝜑 → [𝑡 / 𝑥]𝜓) ↔ (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) → ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜓))))
12	11	bisym-P3.33b.RC 299	. 2 ⊢ ((∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) → ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜓))) ↔ ([𝑡 / 𝑥]𝜑 → [𝑡 / 𝑥]𝜓))
13	8, 12	subimr2-P4.RC 543	1 ⊢ ([𝑡 / 𝑥](𝜑 → 𝜓) → ([𝑡 / 𝑥]𝜑 → [𝑡 / 𝑥]𝜓))

Syntax hints:  term-obj 1   = wff-equals 6  ∀wff-forall 8   → wff-imp 10  [wff-psub 714

This theorem was proved from axioms:  ax-L1 11  ax-L2 12  ax-L3 13  ax-MP 14  ax-GEN 15  ax-L4 16

This theorem depends on definitions:  df-bi-D2.1 107  df-and-D2.2 133  df-true-D2.4 155  df-rcp-AND3 161  df-psub-D6.2 716

This theorem is referenced by:  psubim-P6-L2  790  psubim-P6  791