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Theorem 2exbidv 1861
Description: Formula-building rule for 2 existential quantifiers (deduction form). (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 1818 . 2 (𝜑 → (∃𝑦𝜓 ↔ ∃𝑦𝜒))
32exbidv 1818 1 (𝜑 → (∃𝑥𝑦𝜓 ↔ ∃𝑥𝑦𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wex 1485
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 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-17 1519  ax-ial 1527
This theorem depends on definitions:  df-bi 116
This theorem is referenced by:  3exbidv  1862  4exbidv  1863  cbvex4v  1923  ceqsex3v  2772  ceqsex4v  2773  copsexg  4229  euotd  4239  elopab  4243  elxpi  4627  relop  4761  cbvoprab3  5929  ov6g  5990  th3qlem1  6615  ltresr  7801  fisumcom2  11401  fprodcom2fi  11589
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