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Theorem r2exlem 3088
Description: Lemma factoring out common proof steps in r2exf 3089 an r2ex 3090. Introduced to reduce dependencies on axioms. (Contributed by Wolf Lammen, 10-Jan-2020.)
Hypothesis
Ref Expression
r2exlem.1 (∀𝑥𝐴𝑦𝐵 ¬ 𝜑 ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐵) → ¬ 𝜑))
Assertion
Ref Expression
r2exlem (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ 𝜑))

Proof of Theorem r2exlem
StepHypRef Expression
1 exnal 1794 . . 3 (∃𝑥 ¬ ∀𝑦((𝑥𝐴𝑦𝐵) → ¬ 𝜑) ↔ ¬ ∀𝑥𝑦((𝑥𝐴𝑦𝐵) → ¬ 𝜑))
2 r2exlem.1 . . 3 (∀𝑥𝐴𝑦𝐵 ¬ 𝜑 ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐵) → ¬ 𝜑))
31, 2xchbinxr 324 . 2 (∃𝑥 ¬ ∀𝑦((𝑥𝐴𝑦𝐵) → ¬ 𝜑) ↔ ¬ ∀𝑥𝐴𝑦𝐵 ¬ 𝜑)
4 alinexa 1810 . . . 4 (∀𝑦((𝑥𝐴𝑦𝐵) → ¬ 𝜑) ↔ ¬ ∃𝑦((𝑥𝐴𝑦𝐵) ∧ 𝜑))
54con2bii 346 . . 3 (∃𝑦((𝑥𝐴𝑦𝐵) ∧ 𝜑) ↔ ¬ ∀𝑦((𝑥𝐴𝑦𝐵) → ¬ 𝜑))
65exbii 1814 . 2 (∃𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ 𝜑) ↔ ∃𝑥 ¬ ∀𝑦((𝑥𝐴𝑦𝐵) → ¬ 𝜑))
7 ralnex2 3074 . . 3 (∀𝑥𝐴𝑦𝐵 ¬ 𝜑 ↔ ¬ ∃𝑥𝐴𝑦𝐵 𝜑)
87con2bii 346 . 2 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ¬ ∀𝑥𝐴𝑦𝐵 ¬ 𝜑)
93, 6, 83bitr4ri 293 1 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ 𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  wal 1521  wex 1744  wcel 2030  wral 2941  wrex 2942
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745  df-ral 2946  df-rex 2947
This theorem is referenced by:  r2exf  3089  r2ex  3090
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