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Theorem cbvex 2328
Description: Rule used to change bound variables, using implicit substitution. See cbvexv 2330, cbvexv1 2276, and cbvexvw 1993 for weaker versions. The latter two use fewer axioms. (Contributed by NM, 21-Jun-1993.)
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 1819 . . . 4 𝑦 ¬ 𝜑
3 cbval.2 . . . . 5 𝑥𝜓
43nfn 1819 . . . 4 𝑥 ¬ 𝜓
5 cbval.3 . . . . 5 (𝑥 = 𝑦 → (𝜑𝜓))
65notbid 310 . . . 4 (𝑥 = 𝑦 → (¬ 𝜑 ↔ ¬ 𝜓))
72, 4, 6cbval 2327 . . 3 (∀𝑥 ¬ 𝜑 ↔ ∀𝑦 ¬ 𝜓)
8 alnex 1744 . . 3 (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
9 alnex 1744 . . 3 (∀𝑦 ¬ 𝜓 ↔ ¬ ∃𝑦𝜓)
107, 8, 93bitr3i 293 . 2 (¬ ∃𝑥𝜑 ↔ ¬ ∃𝑦𝜓)
1110con4bii 313 1 (∃𝑥𝜑 ↔ ∃𝑦𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wal 1505  wex 1742  wnf 1746
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-11 2091  ax-12 2104  ax-13 2299
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-ex 1743  df-nf 1747
This theorem is referenced by:  cbvexv  2330  exsbOLD  2420  sb8e  2482  mofOLD  2572  eufOLD  2589  cbveuALT  2632  cbvmoOLD  2633
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