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Theorem cbvexh 1679
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 1568 . . 3 (∃𝑦𝜓 → ∀𝑥𝑦𝜓)
3 cbvexh.1 . . . . 5 (𝜑 → ∀𝑦𝜑)
4 cbvexh.3 . . . . . . 7 (𝑥 = 𝑦 → (𝜑𝜓))
54bicomd 139 . . . . . 6 (𝑥 = 𝑦 → (𝜓𝜑))
65equcoms 1635 . . . . 5 (𝑦 = 𝑥 → (𝜓𝜑))
73, 6equsex 1657 . . . 4 (∃𝑦(𝑦 = 𝑥𝜓) ↔ 𝜑)
8 simpr 108 . . . . 5 ((𝑦 = 𝑥𝜓) → 𝜓)
98eximi 1532 . . . 4 (∃𝑦(𝑦 = 𝑥𝜓) → ∃𝑦𝜓)
107, 9sylbir 133 . . 3 (𝜑 → ∃𝑦𝜓)
112, 10exlimih 1525 . 2 (∃𝑥𝜑 → ∃𝑦𝜓)
123hbex 1568 . . 3 (∃𝑥𝜑 → ∀𝑦𝑥𝜑)
131, 4equsex 1657 . . . 4 (∃𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)
14 simpr 108 . . . . 5 ((𝑥 = 𝑦𝜑) → 𝜑)
1514eximi 1532 . . . 4 (∃𝑥(𝑥 = 𝑦𝜑) → ∃𝑥𝜑)
1613, 15sylbir 133 . . 3 (𝜓 → ∃𝑥𝜑)
1712, 16exlimih 1525 . 2 (∃𝑦𝜓 → ∃𝑥𝜑)
1811, 17impbii 124 1 (∃𝑥𝜑 ↔ ∃𝑦𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wal 1283   = wceq 1285  wex 1422
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  cbvex  1680  sb8eh  1777  cbvexv  1837  euf  1947  mopick  2020
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