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Theorem 2exbidv 1748
 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 1706 . 2 (𝜑 → (∃𝑦𝜓 ↔ ∃𝑦𝜒))
32exbidv 1706 1 (𝜑 → (∃𝑥𝑦𝜓 ↔ ∃𝑥𝑦𝜒))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98  ∃wex 1381 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427 This theorem depends on definitions:  df-bi 110 This theorem is referenced by:  3exbidv  1749  4exbidv  1750  cbvex4v  1805  ceqsex3v  2596  ceqsex4v  2597  copsexg  3981  euotd  3991  elopab  3995  elxpi  4361  relop  4486  cbvoprab3  5580  ov6g  5638  th3qlem1  6208  ltresr  6915
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