Theorem cbvald 2412

Description: Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelim 2456. Usage of this theorem is discouraged because it depends on ax-13 2377. See cbvaldw 2343 for a version with 𝑥, 𝑦 disjoint, not depending on ax-13 2377. (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 1916	. 2 ⊢ Ⅎ𝑥𝜑
2		cbvald.1	. 2 ⊢ Ⅎ𝑦𝜑
3		cbvald.2	. 2 ⊢ (𝜑 → Ⅎ𝑦𝜓)
4		nfvd 1917	. 2 ⊢ (𝜑 → Ⅎ𝑥𝜒)
5		cbvald.3	. 2 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))
6	1, 2, 3, 4, 5	cbv2 2408	1 ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1540  Ⅎwnf 1785

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-11 2163  ax-12 2185  ax-13 2377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ex 1782  df-nf 1786

This theorem is referenced by:  cbvexd  2413  cbvaldva  2414  axextnd  10505  axrepndlem1  10506  axunndlem1  10509  axpowndlem2  10512  axpowndlem3  10513  axpowndlem4  10514  axregndlem2  10517  axregnd  10518  axinfnd  10520  axacndlem5  10525  axacnd  10526  axextdist  35995  distel  35999  wl-sb8eut  37917