Theorem cbvald 2436

Description: Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelim 2480. Usage of this theorem is discouraged because it depends on ax-13 2401. See cbvaldw 2367 for a version with 𝑥, 𝑦 disjoint, not depending on ax-13 2401. (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 1947	. 2 ⊢ Ⅎ𝑥𝜑
2		cbvald.1	. 2 ⊢ Ⅎ𝑦𝜑
3		cbvald.2	. 2 ⊢ (𝜑 → Ⅎ𝑦𝜓)
4		nfvd 1948	. 2 ⊢ (𝜑 → Ⅎ𝑥𝜒)
5		cbvald.3	. 2 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))
6	1, 2, 3, 4, 5	cbv2 2432	1 ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1568  Ⅎwnf 1816

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-11 2194  ax-12 2213  ax-13 2401

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1813  df-nf 1817

This theorem is used by:  cbvexd  2437  cbvaldva  2438  axextnd  10600  axrepndlem1  10601  axunndlem1  10604  axpowndlem2  10607  axpowndlem3  10608  axpowndlem4  10609  axregndlem2  10612  axregnd  10613  axinfnd  10615  axacndlem5  10620  axacnd  10621  axextdist  36376  distel  36380  wl-sb8eut  38341