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Theorem cbvrexdva2 3443
Description: Rule used to change the bound variable in a restricted existential quantifier with implicit substitution which also changes the quantifier domain. Deduction form. (Contributed by David Moews, 1-May-2017.) (Proof shortened by Wolf Lammen, 12-Aug-2023.)
Hypotheses
Ref Expression
cbvraldva2.1 ((𝜑𝑥 = 𝑦) → (𝜓𝜒))
cbvraldva2.2 ((𝜑𝑥 = 𝑦) → 𝐴 = 𝐵)
Assertion
Ref Expression
cbvrexdva2 (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑦𝐵 𝜒))
Distinct variable groups:   𝑦,𝐴   𝜓,𝑦   𝑥,𝐵   𝜒,𝑥   𝜑,𝑥,𝑦
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑦)   𝐴(𝑥)   𝐵(𝑦)

Proof of Theorem cbvrexdva2
StepHypRef Expression
1 simpr 488 . . . . . . . . 9 ((𝜑𝑥 = 𝑦) → 𝑥 = 𝑦)
2 cbvraldva2.2 . . . . . . . . 9 ((𝜑𝑥 = 𝑦) → 𝐴 = 𝐵)
31, 2eleq12d 2910 . . . . . . . 8 ((𝜑𝑥 = 𝑦) → (𝑥𝐴𝑦𝐵))
4 cbvraldva2.1 . . . . . . . 8 ((𝜑𝑥 = 𝑦) → (𝜓𝜒))
53, 4anbi12d 633 . . . . . . 7 ((𝜑𝑥 = 𝑦) → ((𝑥𝐴𝜓) ↔ (𝑦𝐵𝜒)))
65ancoms 462 . . . . . 6 ((𝑥 = 𝑦𝜑) → ((𝑥𝐴𝜓) ↔ (𝑦𝐵𝜒)))
76pm5.32da 582 . . . . 5 (𝑥 = 𝑦 → ((𝜑 ∧ (𝑥𝐴𝜓)) ↔ (𝜑 ∧ (𝑦𝐵𝜒))))
87cbvexvw 2045 . . . 4 (∃𝑥(𝜑 ∧ (𝑥𝐴𝜓)) ↔ ∃𝑦(𝜑 ∧ (𝑦𝐵𝜒)))
9 19.42v 1955 . . . 4 (∃𝑥(𝜑 ∧ (𝑥𝐴𝜓)) ↔ (𝜑 ∧ ∃𝑥(𝑥𝐴𝜓)))
10 19.42v 1955 . . . 4 (∃𝑦(𝜑 ∧ (𝑦𝐵𝜒)) ↔ (𝜑 ∧ ∃𝑦(𝑦𝐵𝜒)))
118, 9, 103bitr3i 304 . . 3 ((𝜑 ∧ ∃𝑥(𝑥𝐴𝜓)) ↔ (𝜑 ∧ ∃𝑦(𝑦𝐵𝜒)))
12 pm5.32 577 . . 3 ((𝜑 → (∃𝑥(𝑥𝐴𝜓) ↔ ∃𝑦(𝑦𝐵𝜒))) ↔ ((𝜑 ∧ ∃𝑥(𝑥𝐴𝜓)) ↔ (𝜑 ∧ ∃𝑦(𝑦𝐵𝜒))))
1311, 12mpbir 234 . 2 (𝜑 → (∃𝑥(𝑥𝐴𝜓) ↔ ∃𝑦(𝑦𝐵𝜒)))
14 df-rex 3139 . 2 (∃𝑥𝐴 𝜓 ↔ ∃𝑥(𝑥𝐴𝜓))
15 df-rex 3139 . 2 (∃𝑦𝐵 𝜒 ↔ ∃𝑦(𝑦𝐵𝜒))
1613, 14, 153bitr4g 317 1 (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑦𝐵 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wex 1781  wcel 2115  wrex 3134
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-cleq 2817  df-clel 2896  df-rex 3139
This theorem is referenced by:  cbvrexdva  3446  mreexexlemd  16911  eulerpartlemgvv  31659  ismnu  40826
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