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Theorem cbvexdh 1817
Description: Deduction used to change bound variables, using implicit substitition, particularly useful in conjunction with dvelim 1909. (Contributed by NM, 2-Jan-2002.) (Proof rewritten by Jim Kingdon, 30-Dec-2017.)
Hypotheses
Ref Expression
cbvexdh.1 (𝜑 → ∀𝑦𝜑)
cbvexdh.2 (𝜑 → (𝜓 → ∀𝑦𝜓))
cbvexdh.3 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
Assertion
Ref Expression
cbvexdh (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒))
Distinct variable groups:   𝜑,𝑥   𝜒,𝑥
Allowed substitution hints:   𝜑(𝑦)   𝜓(𝑥,𝑦)   𝜒(𝑦)

Proof of Theorem cbvexdh
StepHypRef Expression
1 ax-17 1435 . . 3 (𝜑 → ∀𝑥𝜑)
2 ax-17 1435 . . . 4 (𝜒 → ∀𝑥𝜒)
32hbex 1543 . . 3 (∃𝑦𝜒 → ∀𝑥𝑦𝜒)
4 cbvexdh.1 . . . . 5 (𝜑 → ∀𝑦𝜑)
5 cbvexdh.2 . . . . 5 (𝜑 → (𝜓 → ∀𝑦𝜓))
6 cbvexdh.3 . . . . . 6 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
7 equcomi 1608 . . . . . . 7 (𝑦 = 𝑥𝑥 = 𝑦)
8 bicom1 126 . . . . . . 7 ((𝜓𝜒) → (𝜒𝜓))
97, 8imim12i 57 . . . . . 6 ((𝑥 = 𝑦 → (𝜓𝜒)) → (𝑦 = 𝑥 → (𝜒𝜓)))
106, 9syl 14 . . . . 5 (𝜑 → (𝑦 = 𝑥 → (𝜒𝜓)))
114, 5, 10equsexd 1633 . . . 4 (𝜑 → (∃𝑦(𝑦 = 𝑥𝜒) ↔ 𝜓))
12 simpr 107 . . . . 5 ((𝑦 = 𝑥𝜒) → 𝜒)
1312eximi 1507 . . . 4 (∃𝑦(𝑦 = 𝑥𝜒) → ∃𝑦𝜒)
1411, 13syl6bir 157 . . 3 (𝜑 → (𝜓 → ∃𝑦𝜒))
151, 3, 14exlimdh 1503 . 2 (𝜑 → (∃𝑥𝜓 → ∃𝑦𝜒))
161, 5eximdh 1518 . . . 4 (𝜑 → (∃𝑥𝜓 → ∃𝑥𝑦𝜓))
17 19.12 1571 . . . 4 (∃𝑥𝑦𝜓 → ∀𝑦𝑥𝜓)
1816, 17syl6 33 . . 3 (𝜑 → (∃𝑥𝜓 → ∀𝑦𝑥𝜓))
192a1i 9 . . . . 5 (𝜑 → (𝜒 → ∀𝑥𝜒))
201, 19, 6equsexd 1633 . . . 4 (𝜑 → (∃𝑥(𝑥 = 𝑦𝜓) ↔ 𝜒))
21 simpr 107 . . . . 5 ((𝑥 = 𝑦𝜓) → 𝜓)
2221eximi 1507 . . . 4 (∃𝑥(𝑥 = 𝑦𝜓) → ∃𝑥𝜓)
2320, 22syl6bir 157 . . 3 (𝜑 → (𝜒 → ∃𝑥𝜓))
244, 18, 23exlimd2 1502 . 2 (𝜑 → (∃𝑦𝜒 → ∃𝑥𝜓))
2515, 24impbid 124 1 (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  wal 1257   = wceq 1259  wex 1397
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443
This theorem depends on definitions:  df-bi 114
This theorem is referenced by:  cbvexd  1818
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