Theorem chvarv 1907

Description: Implicit substitution of 𝑦 for 𝑥 into a theorem. (Contributed by NM, 20-Apr-1994.)

Hypotheses

Ref	Expression
chv.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
chv.2	⊢ 𝜑

Assertion

Ref	Expression
chvarv	⊢ 𝜓

Distinct variable group:   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑦)

Proof of Theorem chvarv

Step	Hyp	Ref	Expression
1		chv.1	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
2	1	spv 1832	. 2 ⊢ (∀𝑥𝜑 → 𝜓)
3		chv.2	. 2 ⊢ 𝜑
4	2, 3	mpg 1427	1 ⊢ 𝜓

Syntax hints:   → wi 4   ↔ wb 104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514

This theorem depends on definitions:  df-bi 116  df-nf 1437

This theorem is referenced by:  axext3  2120  axsep2  4042  tz6.12f  5443  ltordlem  8237  bdsep2  13073  strcoll2  13170