Theorem psubleq-P6 783

Description: Proper Substitution is Bound to Logical Equivalence.

Hypothesis

Ref	Expression
psubleq-P6.1	⊢ (𝜑 ↔ 𝜓)

Assertion

Ref	Expression
psubleq-P6	⊢ ([𝑡 / 𝑥]𝜑 ↔ [𝑡 / 𝑥]𝜓)

Proof of Theorem psubleq-P6

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-psub-D6.2 716	. 2 ⊢ ([𝑡 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
2		psubleq-P6.1	. . . . . 6 ⊢ (𝜑 ↔ 𝜓)
3	2	subimr-P3.40b.RC 328	. . . . 5 ⊢ ((𝑥 = 𝑦 → 𝜑) ↔ (𝑥 = 𝑦 → 𝜓))
4	3	suballinf-P5 594	. . . 4 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜓))
5	4	subimr-P3.40b.RC 328	. . 3 ⊢ ((𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) ↔ (𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜓)))
6	5	suballinf-P5 594	. 2 ⊢ (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜓)))
7		df-psub-D6.2 716	. . 3 ⊢ ([𝑡 / 𝑥]𝜓 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜓)))
8	7	bisym-P3.33b.RC 299	. 2 ⊢ (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜓)) ↔ [𝑡 / 𝑥]𝜓)
9	1, 6, 8	dbitrns-P4.16.RC 429	1 ⊢ ([𝑡 / 𝑥]𝜑 ↔ [𝑡 / 𝑥]𝜓)

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

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:  psuband-P6  792  psuball2v-P6  796  psubex2v-P6  797  psuball2-P6  798  psubex2-P6  799  psubspliteq-P6  800  psubsplitelof-P6  801