Theorem psubtoisub-P6 765

Description: Conversion from Explicit to Implicit Substitution.

This theorem holds even when '𝑡' contains '𝑥'.

Assertion

Ref	Expression
psubtoisub-P6	⊢ (𝑥 = 𝑡 → (𝜑 ↔ [𝑡 / 𝑥]𝜑))

Proof of Theorem psubtoisub-P6

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfrv-P6 686	. . 3 ⊢ Ⅎ𝑦𝜑
2	1	lemma-L6.05a 764	. 2 ⊢ (𝑥 = 𝑡 → (𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))
3		df-psub-D6.2 716	. . . 4 ⊢ ([𝑡 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
4	3	bisym-P3.33b.RC 299	. . 3 ⊢ (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) ↔ [𝑡 / 𝑥]𝜑)
5	4	subbir-P3.41b.RC 335	. 2 ⊢ ((𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) ↔ (𝜑 ↔ [𝑡 / 𝑥]𝜑))
6	2, 5	subimr2-P4.RC 543	1 ⊢ (𝑥 = 𝑡 → (𝜑 ↔ [𝑡 / 𝑥]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-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:  psubcomp-P6  767  cbvallpsub-P6  768  cbvexpsub-P6  769  psuball2v-P6-L1  795  psubsuccv-P6-L1  805  psubaddv-P6-L1  807  psubmultv-P6-L1  809