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Theorem r2exlem 3262
 Description: Lemma factoring out common proof steps in r2exf 3285 an r2ex 3263. 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 1828 . . 3 (∃𝑥 ¬ ∀𝑦((𝑥𝐴𝑦𝐵) → ¬ 𝜑) ↔ ¬ ∀𝑥𝑦((𝑥𝐴𝑦𝐵) → ¬ 𝜑))
2 r2exlem.1 . . 3 (∀𝑥𝐴𝑦𝐵 ¬ 𝜑 ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐵) → ¬ 𝜑))
31, 2xchbinxr 338 . 2 (∃𝑥 ¬ ∀𝑦((𝑥𝐴𝑦𝐵) → ¬ 𝜑) ↔ ¬ ∀𝑥𝐴𝑦𝐵 ¬ 𝜑)
4 exnalimn 1845 . . 3 (∃𝑦((𝑥𝐴𝑦𝐵) ∧ 𝜑) ↔ ¬ ∀𝑦((𝑥𝐴𝑦𝐵) → ¬ 𝜑))
54exbii 1849 . 2 (∃𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ 𝜑) ↔ ∃𝑥 ¬ ∀𝑦((𝑥𝐴𝑦𝐵) → ¬ 𝜑))
6 ralnex2 3222 . . 3 (∀𝑥𝐴𝑦𝐵 ¬ 𝜑 ↔ ¬ ∃𝑥𝐴𝑦𝐵 𝜑)
76con2bii 361 . 2 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ¬ ∀𝑥𝐴𝑦𝐵 ¬ 𝜑)
83, 5, 73bitr4ri 307 1 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ 𝜑))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536  ∃wex 1781   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-ral 3111  df-rex 3112 This theorem is referenced by:  r2ex  3263  r2exf  3285
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