Theorem axL12-P7 982

Description: ax-L12 29 Derived from Natural Deduction Rules. †

'𝑥' cannot occur in '𝑡'.

Assertion

Ref	Expression
axL12-P7	⊢ (𝑥 = 𝑡 → (𝜑 → ∀𝑥(𝑥 = 𝑡 → 𝜑)))

Distinct variable group:   𝑡,𝑥

Proof of Theorem axL12-P7

Step	Hyp	Ref	Expression
1		ndpsub2-P7.14 839	. . 3 ⊢ (𝑥 = 𝑡 → (𝜑 ↔ [𝑡 / 𝑥]𝜑))
2	1	ndbief-P3.14 179	. 2 ⊢ (𝑥 = 𝑡 → (𝜑 → [𝑡 / 𝑥]𝜑))
3		dfpsubv-P7 977	. . . 4 ⊢ ([𝑡 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑡 → 𝜑))
4	3	subimr-P3.40b.RC 328	. . 3 ⊢ ((𝜑 → [𝑡 / 𝑥]𝜑) ↔ (𝜑 → ∀𝑥(𝑥 = 𝑡 → 𝜑)))
5	4	subimr-P3.40b.RC 328	. 2 ⊢ ((𝑥 = 𝑡 → (𝜑 → [𝑡 / 𝑥]𝜑)) ↔ (𝑥 = 𝑡 → (𝜑 → ∀𝑥(𝑥 = 𝑡 → 𝜑))))
6	2, 5	bimpf-P4.RC 532	1 ⊢ (𝑥 = 𝑡 → (𝜑 → ∀𝑥(𝑥 = 𝑡 → 𝜑)))

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  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)