Theorem cbvexd 2409

Description: Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelim 2452. Usage of this theorem is discouraged because it depends on ax-13 2373. Use the weaker cbvexdw 2342 if possible. (Contributed by NM, 2-Jan-2002.) (Revised by Mario Carneiro, 6-Oct-2016.) (New usage is discouraged.)

Hypotheses

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

Assertion

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

Distinct variable groups:   𝜑,𝑥   𝜒,𝑥

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

Proof of Theorem cbvexd

Step	Hyp	Ref	Expression
1		cbvald.1	. . . 4 ⊢ Ⅎ𝑦𝜑
2		cbvald.2	. . . . 5 ⊢ (𝜑 → Ⅎ𝑦𝜓)
3	2	nfnd 1865	. . . 4 ⊢ (𝜑 → Ⅎ𝑦 ¬ 𝜓)
4		cbvald.3	. . . . 5 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))
5		notbi 322	. . . . 5 ⊢ ((𝜓 ↔ 𝜒) ↔ (¬ 𝜓 ↔ ¬ 𝜒))
6	4, 5	syl6ib 254	. . . 4 ⊢ (𝜑 → (𝑥 = 𝑦 → (¬ 𝜓 ↔ ¬ 𝜒)))
7	1, 3, 6	cbvald 2408	. . 3 ⊢ (𝜑 → (∀𝑥 ¬ 𝜓 ↔ ∀𝑦 ¬ 𝜒))
8		alnex 1788	. . 3 ⊢ (∀𝑥 ¬ 𝜓 ↔ ¬ ∃𝑥𝜓)
9		alnex 1788	. . 3 ⊢ (∀𝑦 ¬ 𝜒 ↔ ¬ ∃𝑦𝜒)
10	7, 8, 9	3bitr3g 316	. 2 ⊢ (𝜑 → (¬ ∃𝑥𝜓 ↔ ¬ ∃𝑦𝜒))
11	10	con4bid 320	1 ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209  ∀wal 1540  ∃wex 1786  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 1975  ax-7 2020  ax-11 2162  ax-12 2179  ax-13 2373

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-ex 1787  df-nf 1791

This theorem is referenced by:  cbvexdva  2411  dfid3  5432  axrepndlem2  10096  axunnd  10099  axpowndlem2  10101  axpownd  10104  axregndlem2  10106  axinfndlem1  10108  axacndlem4  10113  wl-mo2df  35371  wl-eudf  35373