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Theorem cbvexh 1711
Description: Rule used to change bound variables, using implicit substitition. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Feb-2015.)
Hypotheses
Ref Expression
cbvexh.1 (𝜑 → ∀𝑦𝜑)
cbvexh.2 (𝜓 → ∀𝑥𝜓)
cbvexh.3 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvexh (∃𝑥𝜑 ↔ ∃𝑦𝜓)

Proof of Theorem cbvexh
StepHypRef Expression
1 cbvexh.2 . . . 4 (𝜓 → ∀𝑥𝜓)
21hbex 1598 . . 3 (∃𝑦𝜓 → ∀𝑥𝑦𝜓)
3 cbvexh.1 . . . . 5 (𝜑 → ∀𝑦𝜑)
4 cbvexh.3 . . . . . . 7 (𝑥 = 𝑦 → (𝜑𝜓))
54bicomd 140 . . . . . 6 (𝑥 = 𝑦 → (𝜓𝜑))
65equcoms 1667 . . . . 5 (𝑦 = 𝑥 → (𝜓𝜑))
73, 6equsex 1689 . . . 4 (∃𝑦(𝑦 = 𝑥𝜓) ↔ 𝜑)
8 simpr 109 . . . . 5 ((𝑦 = 𝑥𝜓) → 𝜓)
98eximi 1562 . . . 4 (∃𝑦(𝑦 = 𝑥𝜓) → ∃𝑦𝜓)
107, 9sylbir 134 . . 3 (𝜑 → ∃𝑦𝜓)
112, 10exlimih 1555 . 2 (∃𝑥𝜑 → ∃𝑦𝜓)
123hbex 1598 . . 3 (∃𝑥𝜑 → ∀𝑦𝑥𝜑)
131, 4equsex 1689 . . . 4 (∃𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)
14 simpr 109 . . . . 5 ((𝑥 = 𝑦𝜑) → 𝜑)
1514eximi 1562 . . . 4 (∃𝑥(𝑥 = 𝑦𝜑) → ∃𝑥𝜑)
1613, 15sylbir 134 . . 3 (𝜓 → ∃𝑥𝜑)
1712, 16exlimih 1555 . 2 (∃𝑦𝜓 → ∃𝑥𝜑)
1811, 17impbii 125 1 (∃𝑥𝜑 ↔ ∃𝑦𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wal 1312   = wceq 1314  wex 1451
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497
This theorem depends on definitions:  df-bi 116
This theorem is referenced by:  cbvex  1712  sb8eh  1809  cbvexv  1870  euf  1980  mopick  2053
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