Theorem cbvald 2415

Description: Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelim 2459. Usage of this theorem is discouraged because it depends on ax-13 2380. See cbvaldw 2346 for a version with 𝑥, 𝑦 disjoint, not depending on ax-13 2380. (Contributed by NM, 2-Jan-2002.) (Revised by Mario Carneiro, 6-Oct-2016.) (Revised by Wolf Lammen, 13-May-2018.) (New usage is discouraged.)

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 1921	. 2 ⊢ Ⅎ𝑥𝜑
2		cbvald.1	. 2 ⊢ Ⅎ𝑦𝜑
3		cbvald.2	. 2 ⊢ (𝜑 → Ⅎ𝑦𝜓)
4		nfvd 1922	. 2 ⊢ (𝜑 → Ⅎ𝑥𝜒)
5		cbvald.3	. 2 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))
6	1, 2, 3, 4, 5	cbv2 2411	1 ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

Syntax hints:   → wi 4   ↔ wb 207  ∀wal 1545  Ⅎwnf 1790

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-11 2168  ax-12 2189  ax-13 2380

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-ex 1787  df-nf 1791

This theorem is referenced by:  cbvexd  2416  cbvaldva  2417  axextnd  10505  axrepndlem1  10506  axunndlem1  10509  axpowndlem2  10512  axpowndlem3  10513  axpowndlem4  10514  axregndlem2  10517  axregnd  10518  axinfnd  10520  axacndlem5  10525  axacnd  10526  axextdist  36025  distel  36029  wl-sb8eut  37949