Theorem chvarv 1964

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 1882	. 2 ⊢ (∀𝑥𝜑 → 𝜓)
3		chv.2	. 2 ⊢ 𝜑
4	2, 3	mpg 1473	1 ⊢ 𝜓

Syntax hints:   → wi 4   ↔ wb 105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1469  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556

This theorem depends on definitions:  df-bi 117  df-nf 1483

This theorem is referenced by:  axext3  2187  axsep2  4162  tz6.12f  5599  ltordlem  8537  bdsep2  15686  strcoll2  15783