Theorem psubthm-P6 786

Description: Proper Substitution Applied to a Theorem.

Hypothesis

Ref	Expression
psubthm-P6.1	⊢ 𝜑

Assertion

Ref	Expression
psubthm-P6	⊢ [𝑡 / 𝑥]𝜑

Proof of Theorem psubthm-P6

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		psubthm-P6.1	. . . . . 6 ⊢ 𝜑
2	1	rcp-NDIMP0addall 207	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝜑)
3	2	ax-GEN 15	. . . 4 ⊢ ∀𝑥(𝑥 = 𝑦 → 𝜑)
4	3	rcp-NDIMP0addall 207	. . 3 ⊢ (𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑))
5	4	ax-GEN 15	. 2 ⊢ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑))
6		df-psub-D6.2 716	. 2 ⊢ ([𝑡 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
7	5, 6	bimpr-P4.RC 534	1 ⊢ [𝑡 / 𝑥]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

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

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