Theorem psubtoisubv-P6 725

Description: Conversion From Explicit to Implicit Substitution (for when '𝑡' does not contain '𝑥').

'𝑥' cannot occur in '𝑡'.

The full form is proved later as psubtoisub-P6 765.

Assertion

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

Distinct variable group:   𝑡,𝑥

Proof of Theorem psubtoisubv-P6

Step	Hyp	Ref	Expression
1		lemma-L6.01a 724	. 2 ⊢ (𝑥 = 𝑡 → (𝜑 ↔ ∀𝑥(𝑥 = 𝑡 → 𝜑)))
2		dfpsubv-P6 717	. . . 4 ⊢ ([𝑡 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑡 → 𝜑))
3	2	bisym-P3.33b.RC 299	. . 3 ⊢ (∀𝑥(𝑥 = 𝑡 → 𝜑) ↔ [𝑡 / 𝑥]𝜑)
4	3	subbir-P3.41b.RC 335	. 2 ⊢ ((𝜑 ↔ ∀𝑥(𝑥 = 𝑡 → 𝜑)) ↔ (𝜑 ↔ [𝑡 / 𝑥]𝜑))
5	1, 4	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-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:  ndpsub2-P7.14  839