Theorem cbvex 2390

Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2363. Check out cbvexvw 2032, cbvexv1 2330 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 1852	. . . 4 ⊢ Ⅎ𝑦 ¬ 𝜑
3		cbval.2	. . . . 5 ⊢ Ⅎ𝑥𝜓
4	3	nfn 1852	. . . 4 ⊢ Ⅎ𝑥 ¬ 𝜓
5		cbval.3	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
6	5	notbid 318	. . . 4 ⊢ (𝑥 = 𝑦 → (¬ 𝜑 ↔ ¬ 𝜓))
7	2, 4, 6	cbval 2389	. . 3 ⊢ (∀𝑥 ¬ 𝜑 ↔ ∀𝑦 ¬ 𝜓)
8		alnex 1775	. . 3 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
9		alnex 1775	. . 3 ⊢ (∀𝑦 ¬ 𝜓 ↔ ¬ ∃𝑦𝜓)
10	7, 8, 9	3bitr3i 301	. 2 ⊢ (¬ ∃𝑥𝜑 ↔ ¬ ∃𝑦𝜓)
11	10	con4bii 321	1 ⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205  ∀wal 1531  ∃wex 1773  Ⅎwnf 1777

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-11 2146  ax-12 2163  ax-13 2363

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-ex 1774  df-nf 1778

This theorem is referenced by:  cbvexv  2392  sb8e  2509  cbveuALT  2596