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Theorem cbvrexdva2 2711
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.)
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 110 . . . . 5 ((𝜑𝑥 = 𝑦) → 𝑥 = 𝑦)
2 cbvraldva2.2 . . . . 5 ((𝜑𝑥 = 𝑦) → 𝐴 = 𝐵)
31, 2eleq12d 2248 . . . 4 ((𝜑𝑥 = 𝑦) → (𝑥𝐴𝑦𝐵))
4 cbvraldva2.1 . . . 4 ((𝜑𝑥 = 𝑦) → (𝜓𝜒))
53, 4anbi12d 473 . . 3 ((𝜑𝑥 = 𝑦) → ((𝑥𝐴𝜓) ↔ (𝑦𝐵𝜒)))
65cbvexdva 1929 . 2 (𝜑 → (∃𝑥(𝑥𝐴𝜓) ↔ ∃𝑦(𝑦𝐵𝜒)))
7 df-rex 2461 . 2 (∃𝑥𝐴 𝜓 ↔ ∃𝑥(𝑥𝐴𝜓))
8 df-rex 2461 . 2 (∃𝑦𝐵 𝜒 ↔ ∃𝑦(𝑦𝐵𝜒))
96, 7, 83bitr4g 223 1 (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑦𝐵 𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wex 1492  wcel 2148  wrex 2456
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-cleq 2170  df-clel 2173  df-rex 2461
This theorem is referenced by:  cbvrexdva  2713  acexmid  5867
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