Theorem cbvex 2398

Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2371. Check out cbvexvw 2045, cbvexv1 2342 for weaker versions requiring fewer axioms. (Contributed by NM, 21-Jun-1993.) (New usage is discouraged.)

Hypotheses

Ref	Expression
cbval.1	⊢ Ⅎ𝑦𝜑
cbval.2	⊢ Ⅎ𝑥𝜓
cbval.3	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
cbvex	⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜓)

Proof of Theorem cbvex

Step	Hyp	Ref	Expression
1		cbval.1	. . . . 5 ⊢ Ⅎ𝑦𝜑
2	1	nfn 1865	. . . 4 ⊢ Ⅎ𝑦 ¬ 𝜑
3		cbval.2	. . . . 5 ⊢ Ⅎ𝑥𝜓
4	3	nfn 1865	. . . 4 ⊢ Ⅎ𝑥 ¬ 𝜓
5		cbval.3	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
6	5	notbid 321	. . . 4 ⊢ (𝑥 = 𝑦 → (¬ 𝜑 ↔ ¬ 𝜓))
7	2, 4, 6	cbval 2397	. . 3 ⊢ (∀𝑥 ¬ 𝜑 ↔ ∀𝑦 ¬ 𝜓)
8		alnex 1789	. . 3 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
9		alnex 1789	. . 3 ⊢ (∀𝑦 ¬ 𝜓 ↔ ¬ ∃𝑦𝜓)
10	7, 8, 9	3bitr3i 304	. 2 ⊢ (¬ ∃𝑥𝜑 ↔ ¬ ∃𝑦𝜓)
11	10	con4bii 324	1 ⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209  ∀wal 1541  ∃wex 1787  Ⅎwnf 1791

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-11 2158  ax-12 2175  ax-13 2371

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-ex 1788  df-nf 1792

This theorem is referenced by:  cbvexv  2400  sb8e  2521  cbveuALT  2609