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Theorem 2rexbidv 2364
Description: Formula-building rule for restricted existential quantifiers (deduction rule). (Contributed by NM, 28-Jan-2006.)
Hypothesis
Ref Expression
2ralbidv.1 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
2rexbidv (𝜑 → (∃𝑥𝐴𝑦𝐵 𝜓 ↔ ∃𝑥𝐴𝑦𝐵 𝜒))
Distinct variable groups:   𝜑,𝑥   𝜑,𝑦
Allowed substitution hints:   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦)   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)

Proof of Theorem 2rexbidv
StepHypRef Expression
1 2ralbidv.1 . . 3 (𝜑 → (𝜓𝜒))
21rexbidv 2342 . 2 (𝜑 → (∃𝑦𝐵 𝜓 ↔ ∃𝑦𝐵 𝜒))
32rexbidv 2342 1 (𝜑 → (∃𝑥𝐴𝑦𝐵 𝜓 ↔ ∃𝑥𝐴𝑦𝐵 𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102  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-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-4 1414  ax-17 1433  ax-ial 1441
This theorem depends on definitions:  df-bi 114  df-nf 1364  df-rex 2327
This theorem is referenced by:  f1oiso  5490  elrnmpt2g  5638  elrnmpt2  5639  ralrnmpt2  5640  rexrnmpt2  5641  ovelrn  5674  eroveu  6225  genipv  6635  genpelxp  6637  genpelvl  6638  genpelvu  6639  axcnre  6983  apreap  7622  apreim  7638
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