Theorem cbvexdw 2373

Description: Deduction used to change bound variables, using implicit substitution. Version of cbvexd 2442 with a disjoint variable condition, which does not require ax-13 2406. (Contributed by NM, 2-Jan-2002.) Avoid ax-13 2406. (Revised by GG, 10-Jan-2024.)

Hypotheses

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

Assertion

Ref	Expression
cbvexdw	⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒))

Distinct variable groups:   𝜑,𝑥   𝜒,𝑥   𝑥,𝑦

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

Proof of Theorem cbvexdw

Step	Hyp	Ref	Expression
1		cbvaldw.1	. . . 4 ⊢ Ⅎ𝑦𝜑
2		cbvaldw.2	. . . . 5 ⊢ (𝜑 → Ⅎ𝑦𝜓)
3	2	nfnd 1891	. . . 4 ⊢ (𝜑 → Ⅎ𝑦 ¬ 𝜓)
4		cbvaldw.3	. . . . 5 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))
5		notbi 322	. . . . 5 ⊢ ((𝜓 ↔ 𝜒) ↔ (¬ 𝜓 ↔ ¬ 𝜒))
6	4, 5	imbitrdi 254	. . . 4 ⊢ (𝜑 → (𝑥 = 𝑦 → (¬ 𝜓 ↔ ¬ 𝜒)))
7	1, 3, 6	cbvaldw 2372	. . 3 ⊢ (𝜑 → (∀𝑥 ¬ 𝜓 ↔ ∀𝑦 ¬ 𝜒))
8		alnex 1814	. . 3 ⊢ (∀𝑥 ¬ 𝜓 ↔ ¬ ∃𝑥𝜓)
9		alnex 1814	. . 3 ⊢ (∀𝑦 ¬ 𝜒 ↔ ¬ ∃𝑦𝜒)
10	7, 8, 9	3bitr3g 316	. 2 ⊢ (𝜑 → (¬ ∃𝑥𝜓 ↔ ¬ ∃𝑦𝜒))
11	10	con4bid 320	1 ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209  ∀wal 1568  ∃wex 1812  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 2195  ax-12 2216

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:  axtcond  37048  bj-gabima  37635  bj-opabco  37891  wl-mo2t  38289