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Theorem rexbidv2 2493
Description: Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 22-May-1999.)
Hypothesis
Ref Expression
rexbidv2.1 (𝜑 → ((𝑥𝐴𝜓) ↔ (𝑥𝐵𝜒)))
Assertion
Ref Expression
rexbidv2 (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐵 𝜒))
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem rexbidv2
StepHypRef Expression
1 rexbidv2.1 . . 3 (𝜑 → ((𝑥𝐴𝜓) ↔ (𝑥𝐵𝜒)))
21exbidv 1836 . 2 (𝜑 → (∃𝑥(𝑥𝐴𝜓) ↔ ∃𝑥(𝑥𝐵𝜒)))
3 df-rex 2474 . 2 (∃𝑥𝐴 𝜓 ↔ ∃𝑥(𝑥𝐴𝜓))
4 df-rex 2474 . 2 (∃𝑥𝐵 𝜒 ↔ ∃𝑥(𝑥𝐵𝜒))
52, 3, 43bitr4g 223 1 (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐵 𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wex 1503  wcel 2160  wrex 2469
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 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-17 1537  ax-ial 1545
This theorem depends on definitions:  df-bi 117  df-rex 2474
This theorem is referenced by:  rexss  3237  rexsupp  5661  isoini  5840  elfi2  7001  ltexpi  7366  rexuz  9610  4sqexercise2  12431  4sqlemsdc  12432  sscoll2  15198
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