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Theorem cbvrexdva2 2553
 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 107 . . . . 5 ((𝜑𝑥 = 𝑦) → 𝑥 = 𝑦)
2 cbvraldva2.2 . . . . 5 ((𝜑𝑥 = 𝑦) → 𝐴 = 𝐵)
31, 2eleq12d 2122 . . . 4 ((𝜑𝑥 = 𝑦) → (𝑥𝐴𝑦𝐵))
4 cbvraldva2.1 . . . 4 ((𝜑𝑥 = 𝑦) → (𝜓𝜒))
53, 4anbi12d 450 . . 3 ((𝜑𝑥 = 𝑦) → ((𝑥𝐴𝜓) ↔ (𝑦𝐵𝜒)))
65cbvexdva 1818 . 2 (𝜑 → (∃𝑥(𝑥𝐴𝜓) ↔ ∃𝑦(𝑦𝐵𝜒)))
7 df-rex 2327 . 2 (∃𝑥𝐴 𝜓 ↔ ∃𝑥(𝑥𝐴𝜓))
8 df-rex 2327 . 2 (∃𝑦𝐵 𝜒 ↔ ∃𝑦(𝑦𝐵𝜒))
96, 7, 83bitr4g 216 1 (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑦𝐵 𝜒))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 101   ↔ wb 102   = wceq 1257  ∃wex 1395   ∈ wcel 1407  ∃wrex 2322 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 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-4 1414  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-ext 2036 This theorem depends on definitions:  df-bi 114  df-nf 1364  df-cleq 2047  df-clel 2050  df-rex 2327 This theorem is referenced by:  cbvrexdva  2555  acexmid  5536
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