Theorem dfpsub-P7 978

Description: df-psub-D6.2 716, Derived from Natural Deduction Rules. †

Assertion

Ref	Expression
dfpsub-P7	⊢ ([𝑡 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))

Distinct variable groups:   𝜑,𝑦   𝑡,𝑦   𝑥,𝑦

Proof of Theorem dfpsub-P7

Step	Hyp	Ref	Expression
1		ndpsub4-P7.16 841	. 2 ⊢ ([𝑡 / 𝑥]𝜑 ↔ [𝑡 / 𝑦][𝑦 / 𝑥]𝜑)
2		dfpsubv-P7 977	. 2 ⊢ ([𝑡 / 𝑦][𝑦 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → [𝑦 / 𝑥]𝜑))
3		dfpsubv-P7 977	. . . 4 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
4	3	subimr-P3.40b.RC 328	. . 3 ⊢ ((𝑦 = 𝑡 → [𝑦 / 𝑥]𝜑) ↔ (𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
5	4	suball-P7.RC 974	. 2 ⊢ (∀𝑦(𝑦 = 𝑡 → [𝑦 / 𝑥]𝜑) ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
6	1, 2, 5	dbitrns-P4.16.RC 429	1 ⊢ ([𝑡 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))

Syntax hints:  term-obj 1   = wff-equals 6  ∀wff-forall 8   → wff-imp 10   ↔ wff-bi 104  [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  ax-L5 17  ax-L6 18  ax-L7 19  ax-L10 27  ax-L11 28  ax-L12 29

This theorem depends on definitions:  df-bi-D2.1 107  df-and-D2.2 133  df-or-D2.3 145  df-true-D2.4 155  df-rcp-AND3 161  df-exists-D5.1 596  df-nfree-D6.1 682  df-psub-D6.2 716

This theorem is referenced by: (None)