Theorem chvar 1781

Description: Implicit substitution of 𝑦 for 𝑥 into a theorem. (Contributed by Raph Levien, 9-Jul-2003.) (Revised by Mario Carneiro, 3-Oct-2016.)

Hypotheses

Ref	Expression
chvar.1	⊢ Ⅎ𝑥𝜓
chvar.2	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
chvar.3	⊢ 𝜑

Assertion

Ref	Expression
chvar	⊢ 𝜓

Proof of Theorem chvar

Step	Hyp	Ref	Expression
1		chvar.1	. . 3 ⊢ Ⅎ𝑥𝜓
2		chvar.2	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
3	2	biimpd 144	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
4	1, 3	spim 1762	. 2 ⊢ (∀𝑥𝜑 → 𝜓)
5		chvar.3	. 2 ⊢ 𝜑
6	4, 5	mpg 1475	1 ⊢ 𝜓

Syntax hints:   → wi 4   ↔ wb 105  Ⅎwnf 1484

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 1471  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-4 1534  ax-i9 1554  ax-ial 1558

This theorem depends on definitions:  df-bi 117  df-nf 1485

This theorem is referenced by:  csbhypf  3133  opelopabsb  4310  findes  4655  fvmptssdm  5671  dfoprab4f  6286  dom2lem  6870  uzind4s  9718  fsumsplitf  11763