Theorem chvar 2409

Description: Implicit substitution of 𝑦 for 𝑥 into a theorem. Usage of this theorem is discouraged because it depends on ax-13 2386. Use the weaker chvarfv 2238 if possible. (Contributed by Raph Levien, 9-Jul-2003.) (Revised by Mario Carneiro, 3-Oct-2016.) (New usage is discouraged.)

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 231	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
4	1, 3	spim 2401	. 2 ⊢ (∀𝑥𝜑 → 𝜓)
5		chvar.3	. 2 ⊢ 𝜑
6	4, 5	mpg 1794	1 ⊢ 𝜓

Syntax hints:   → wi 4   ↔ wb 208  Ⅎwnf 1780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-12 2173  ax-13 2386

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1777  df-nf 1781

This theorem is referenced by:  chvarv  2410  zfcndrep  10030