Theorem cbvex 2407

Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2380. Check out cbvexvw 2044, cbvexv1 2350 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 1864	. . . 4 ⊢ Ⅎ𝑦 ¬ 𝜑
3		cbval.2	. . . . 5 ⊢ Ⅎ𝑥𝜓
4	3	nfn 1864	. . . 4 ⊢ Ⅎ𝑥 ¬ 𝜓
5		cbval.3	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
6	5	notbid 319	. . . 4 ⊢ (𝑥 = 𝑦 → (¬ 𝜑 ↔ ¬ 𝜓))
7	2, 4, 6	cbval 2406	. . 3 ⊢ (∀𝑥 ¬ 𝜑 ↔ ∀𝑦 ¬ 𝜓)
8		alnex 1788	. . 3 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
9		alnex 1788	. . 3 ⊢ (∀𝑦 ¬ 𝜓 ↔ ¬ ∃𝑦𝜓)
10	7, 8, 9	3bitr3i 302	. 2 ⊢ (¬ ∃𝑥𝜑 ↔ ¬ ∃𝑦𝜓)
11	10	con4bii 322	1 ⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207  ∀wal 1545  ∃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 1974  ax-7 2015  ax-11 2168  ax-12 2189  ax-13 2380

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-ex 1787  df-nf 1791

This theorem is referenced by:  cbvexv  2409  sb8e  2526  cbveuALT  2612