Theorem chvarv 1812

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 1740	. 2 ⊢ (∀𝑥𝜑 → 𝜓)
3		chv.2	. 2 ⊢ 𝜑
4	2, 3	mpg 1340	1 ⊢ 𝜓

Syntax hints:   → wi 4   ↔ wb 98

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427

This theorem depends on definitions:  df-bi 110  df-nf 1350

This theorem is referenced by:  axext3  2023  axsep2  3876  tz6.12f  5202  tfrlem3-2  5927  bdsep2  10006  strcoll2  10108