Theorem cbvald 2427

Description: Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelim 2472. (Contributed by NM, 2-Jan-2002.) (Revised by Mario Carneiro, 6-Oct-2016.) (Revised by Wolf Lammen, 13-May-2018.)

Hypotheses

Ref	Expression
cbvald.1	⊢ Ⅎ𝑦𝜑
cbvald.2	⊢ (𝜑 → Ⅎ𝑦𝜓)
cbvald.3	⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))

Assertion

Ref	Expression
cbvald	⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

Distinct variable groups:   𝜑,𝑥   𝜒,𝑥

Allowed substitution hints:   𝜑(𝑦)   𝜓(𝑥,𝑦)   𝜒(𝑦)

Proof of Theorem cbvald

Step	Hyp	Ref	Expression
1		nfv 2013	. 2 ⊢ Ⅎ𝑥𝜑
2		cbvald.1	. 2 ⊢ Ⅎ𝑦𝜑
3		cbvald.2	. 2 ⊢ (𝜑 → Ⅎ𝑦𝜓)
4		nfvd 2014	. 2 ⊢ (𝜑 → Ⅎ𝑥𝜒)
5		cbvald.3	. 2 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))
6	1, 2, 3, 4, 5	cbv2 2422	1 ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

Syntax hints:   → wi 4   ↔ wb 198  ∀wal 1654  Ⅎwnf 1882

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-ex 1879  df-nf 1883

This theorem is referenced by:  cbvexd  2428  axextnd  9735  axrepndlem1  9736  axunndlem1  9739  axpowndlem2  9742  axpowndlem3  9743  axpowndlem4  9744  axregndlem2  9747  axregnd  9748  axinfnd  9750  axacndlem5  9755  axacnd  9756  axextdist  32238  distel  32242  wl-sb8eut  33898