Theorem chvarv 1853

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 1781	. 2 ⊢ (∀𝑥𝜑 → 𝜓)
3		chv.2	. 2 ⊢ 𝜑
4	2, 3	mpg 1380	1 ⊢ 𝜓

Syntax hints:   → wi 4   ↔ wb 103

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467

This theorem depends on definitions:  df-bi 115  df-nf 1390

This theorem is referenced by:  axext3  2064  axsep2  3897  tz6.12f  5223  tfrlem3-2  5950  bdsep2  10677  strcoll2  10778