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Theorem cbvex 2406
 Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2379. Check out cbvexvw 2044, cbvexv1 2351 for weaker versions requiring fewer axioms. (Contributed by NM, 21-Jun-1993.) (New usage is discouraged.)
Hypotheses
Ref Expression
cbval.1 𝑦𝜑
cbval.2 𝑥𝜓
cbval.3 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvex (∃𝑥𝜑 ↔ ∃𝑦𝜓)

Proof of Theorem cbvex
StepHypRef Expression
1 cbval.1 . . . . 5 𝑦𝜑
21nfn 1858 . . . 4 𝑦 ¬ 𝜑
3 cbval.2 . . . . 5 𝑥𝜓
43nfn 1858 . . . 4 𝑥 ¬ 𝜓
5 cbval.3 . . . . 5 (𝑥 = 𝑦 → (𝜑𝜓))
65notbid 321 . . . 4 (𝑥 = 𝑦 → (¬ 𝜑 ↔ ¬ 𝜓))
72, 4, 6cbval 2405 . . 3 (∀𝑥 ¬ 𝜑 ↔ ∀𝑦 ¬ 𝜓)
8 alnex 1783 . . 3 (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
9 alnex 1783 . . 3 (∀𝑦 ¬ 𝜓 ↔ ¬ ∃𝑦𝜓)
107, 8, 93bitr3i 304 . 2 (¬ ∃𝑥𝜑 ↔ ¬ ∃𝑦𝜓)
1110con4bii 324 1 (∃𝑥𝜑 ↔ ∃𝑦𝜓)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209  ∀wal 1536  ∃wex 1781  Ⅎwnf 1785 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-11 2158  ax-12 2175  ax-13 2379 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786 This theorem is referenced by:  cbvexv  2408  sb8e  2537  cbveuALT  2669
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