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Theorem cbvexh 1685
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 1572 . . 3 (∃𝑦𝜓 → ∀𝑥𝑦𝜓)
3 cbvexh.1 . . . . 5 (𝜑 → ∀𝑦𝜑)
4 cbvexh.3 . . . . . . 7 (𝑥 = 𝑦 → (𝜑𝜓))
54bicomd 139 . . . . . 6 (𝑥 = 𝑦 → (𝜓𝜑))
65equcoms 1641 . . . . 5 (𝑦 = 𝑥 → (𝜓𝜑))
73, 6equsex 1663 . . . 4 (∃𝑦(𝑦 = 𝑥𝜓) ↔ 𝜑)
8 simpr 108 . . . . 5 ((𝑦 = 𝑥𝜓) → 𝜓)
98eximi 1536 . . . 4 (∃𝑦(𝑦 = 𝑥𝜓) → ∃𝑦𝜓)
107, 9sylbir 133 . . 3 (𝜑 → ∃𝑦𝜓)
112, 10exlimih 1529 . 2 (∃𝑥𝜑 → ∃𝑦𝜓)
123hbex 1572 . . 3 (∃𝑥𝜑 → ∀𝑦𝑥𝜑)
131, 4equsex 1663 . . . 4 (∃𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)
14 simpr 108 . . . . 5 ((𝑥 = 𝑦𝜑) → 𝜑)
1514eximi 1536 . . . 4 (∃𝑥(𝑥 = 𝑦𝜑) → ∃𝑥𝜑)
1613, 15sylbir 133 . . 3 (𝜓 → ∃𝑥𝜑)
1712, 16exlimih 1529 . 2 (∃𝑦𝜓 → ∃𝑥𝜑)
1811, 17impbii 124 1 (∃𝑥𝜑 ↔ ∃𝑦𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wal 1287   = wceq 1289  wex 1426
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 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  cbvex  1686  sb8eh  1783  cbvexv  1843  euf  1953  mopick  2026
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