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Theorem 2reu5lem2 3412
Description: Lemma for 2reu5 3414. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
Assertion
Ref Expression
2reu5lem2 (∀𝑥𝐴 ∃*𝑦𝐵 𝜑 ↔ ∀𝑥∃*𝑦(𝑥𝐴𝑦𝐵𝜑))
Distinct variable groups:   𝑦,𝐴   𝑥,𝐵   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑦)

Proof of Theorem 2reu5lem2
StepHypRef Expression
1 df-rmo 2919 . . 3 (∃*𝑦𝐵 𝜑 ↔ ∃*𝑦(𝑦𝐵𝜑))
21ralbii 2979 . 2 (∀𝑥𝐴 ∃*𝑦𝐵 𝜑 ↔ ∀𝑥𝐴 ∃*𝑦(𝑦𝐵𝜑))
3 df-ral 2916 . . 3 (∀𝑥𝐴 ∃*𝑦(𝑦𝐵𝜑) ↔ ∀𝑥(𝑥𝐴 → ∃*𝑦(𝑦𝐵𝜑)))
4 moanimv 2530 . . . . . 6 (∃*𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)) ↔ (𝑥𝐴 → ∃*𝑦(𝑦𝐵𝜑)))
54bicomi 214 . . . . 5 ((𝑥𝐴 → ∃*𝑦(𝑦𝐵𝜑)) ↔ ∃*𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)))
6 3anass 1041 . . . . . . 7 ((𝑥𝐴𝑦𝐵𝜑) ↔ (𝑥𝐴 ∧ (𝑦𝐵𝜑)))
76bicomi 214 . . . . . 6 ((𝑥𝐴 ∧ (𝑦𝐵𝜑)) ↔ (𝑥𝐴𝑦𝐵𝜑))
87mobii 2492 . . . . 5 (∃*𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)) ↔ ∃*𝑦(𝑥𝐴𝑦𝐵𝜑))
95, 8bitri 264 . . . 4 ((𝑥𝐴 → ∃*𝑦(𝑦𝐵𝜑)) ↔ ∃*𝑦(𝑥𝐴𝑦𝐵𝜑))
109albii 1746 . . 3 (∀𝑥(𝑥𝐴 → ∃*𝑦(𝑦𝐵𝜑)) ↔ ∀𝑥∃*𝑦(𝑥𝐴𝑦𝐵𝜑))
113, 10bitri 264 . 2 (∀𝑥𝐴 ∃*𝑦(𝑦𝐵𝜑) ↔ ∀𝑥∃*𝑦(𝑥𝐴𝑦𝐵𝜑))
122, 11bitri 264 1 (∀𝑥𝐴 ∃*𝑦𝐵 𝜑 ↔ ∀𝑥∃*𝑦(𝑥𝐴𝑦𝐵𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1037  wal 1480  wcel 1989  ∃*wmo 2470  wral 2911  ∃*wrmo 2914
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-10 2018  ax-12 2046
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-eu 2473  df-mo 2474  df-ral 2916  df-rmo 2919
This theorem is referenced by:  2reu5lem3  3413
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