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Theorem r2exlem 3138
Description: Lemma factoring out common proof steps in r2exf 3274 an r2ex 3190. 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 1822 . . 3 (∃𝑥 ¬ ∀𝑦((𝑥𝐴𝑦𝐵) → ¬ 𝜑) ↔ ¬ ∀𝑥𝑦((𝑥𝐴𝑦𝐵) → ¬ 𝜑))
2 r2exlem.1 . . 3 (∀𝑥𝐴𝑦𝐵 ¬ 𝜑 ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐵) → ¬ 𝜑))
31, 2xchbinxr 335 . 2 (∃𝑥 ¬ ∀𝑦((𝑥𝐴𝑦𝐵) → ¬ 𝜑) ↔ ¬ ∀𝑥𝐴𝑦𝐵 ¬ 𝜑)
4 exnalimn 1839 . . 3 (∃𝑦((𝑥𝐴𝑦𝐵) ∧ 𝜑) ↔ ¬ ∀𝑦((𝑥𝐴𝑦𝐵) → ¬ 𝜑))
54exbii 1843 . 2 (∃𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ 𝜑) ↔ ∃𝑥 ¬ ∀𝑦((𝑥𝐴𝑦𝐵) → ¬ 𝜑))
6 ralnex2 3128 . . 3 (∀𝑥𝐴𝑦𝐵 ¬ 𝜑 ↔ ¬ ∃𝑥𝐴𝑦𝐵 𝜑)
76con2bii 357 . 2 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ¬ ∀𝑥𝐴𝑦𝐵 ¬ 𝜑)
83, 5, 73bitr4ri 304 1 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ 𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395  wal 1532  wex 1774  wcel 2099  wral 3056  wrex 3065
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804
This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1775  df-ral 3057  df-rex 3066
This theorem is referenced by:  r2ex  3190  r2exf  3274
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