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Theorem 2exbidv 1764
Description: Formula-building rule for 2 existential quantifiers (deduction rule). (Contributed by NM, 1-May-1995.)
Hypothesis
Ref Expression
2albidv.1 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
2exbidv (𝜑 → (∃𝑥𝑦𝜓 ↔ ∃𝑥𝑦𝜒))
Distinct variable groups:   𝜑,𝑥   𝜑,𝑦
Allowed substitution hints:   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦)

Proof of Theorem 2exbidv
StepHypRef Expression
1 2albidv.1 . . 3 (𝜑 → (𝜓𝜒))
21exbidv 1722 . 2 (𝜑 → (∃𝑦𝜓 ↔ ∃𝑦𝜒))
32exbidv 1722 1 (𝜑 → (∃𝑥𝑦𝜓 ↔ ∃𝑥𝑦𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102  wex 1397
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 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-17 1435  ax-ial 1443
This theorem depends on definitions:  df-bi 114
This theorem is referenced by:  3exbidv  1765  4exbidv  1766  cbvex4v  1821  ceqsex3v  2613  ceqsex4v  2614  copsexg  4009  euotd  4019  elopab  4023  elxpi  4389  relop  4514  cbvoprab3  5608  ov6g  5666  th3qlem1  6239  ltresr  6973
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