Theorem chvarv 2401

Description: Implicit substitution of 𝑦 for 𝑥 into a theorem. Usage of this theorem is discouraged because it depends on ax-13 2377. Use the weaker chvarvv 1998 if possible. (Contributed by NM, 20-Apr-1994.) (Proof shortened by Wolf Lammen, 22-Apr-2018.) (New usage is discouraged.)

Hypotheses

Ref	Expression
chvarv.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
chvarv.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		nfv 1914	. 2 ⊢ Ⅎ𝑥𝜓
2		chvarv.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
3		chvarv.2	. 2 ⊢ 𝜑
4	1, 2, 3	chvar 2400	1 ⊢ 𝜓

Syntax hints:   → wi 4   ↔ wb 206

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-12 2177  ax-13 2377

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-nf 1784

This theorem is referenced by: (None)