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Theorem rexbii2 2468
 Description: Inference adding different restricted existential quantifiers to each side of an equivalence. (Contributed by NM, 4-Feb-2004.)
Hypothesis
Ref Expression
rexbii2.1 ((𝑥𝐴𝜑) ↔ (𝑥𝐵𝜓))
Assertion
Ref Expression
rexbii2 (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐵 𝜓)

Proof of Theorem rexbii2
StepHypRef Expression
1 rexbii2.1 . . 3 ((𝑥𝐴𝜑) ↔ (𝑥𝐵𝜓))
21exbii 1585 . 2 (∃𝑥(𝑥𝐴𝜑) ↔ ∃𝑥(𝑥𝐵𝜓))
3 df-rex 2441 . 2 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
4 df-rex 2441 . 2 (∃𝑥𝐵 𝜓 ↔ ∃𝑥(𝑥𝐵𝜓))
52, 3, 43bitr4i 211 1 (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐵 𝜓)
 Colors of variables: wff set class Syntax hints:   ∧ wa 103   ↔ wb 104  ∃wex 1472   ∈ wcel 2128  ∃wrex 2436 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1427  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-4 1490  ax-ial 1514 This theorem depends on definitions:  df-bi 116  df-rex 2441 This theorem is referenced by:  rexeqbii  2470  rexbiia  2472  rexrab  2875  rexdifpr  3588  rexdifsn  3691  bnd2  4133  suplocsrlemb  7709  rexuz2  9475  rexrp  9565  rexuz3  10872
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