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Theorem 2reu5lem2 3660
Description: Lemma for 2reu5 3662. (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,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑥,𝑦)

Proof of Theorem 2reu5lem2
StepHypRef Expression
1 df-rmo 3062 . . 3 (∃*𝑦𝐵 𝜑 ↔ ∃*𝑦(𝑦𝐵𝜑))
21ralbii 3081 . 2 (∀𝑥𝐴 ∃*𝑦𝐵 𝜑 ↔ ∀𝑥𝐴 ∃*𝑦(𝑦𝐵𝜑))
3 df-ral 3059 . . 3 (∀𝑥𝐴 ∃*𝑦(𝑦𝐵𝜑) ↔ ∀𝑥(𝑥𝐴 → ∃*𝑦(𝑦𝐵𝜑)))
4 moanimv 2623 . . . . . 6 (∃*𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)) ↔ (𝑥𝐴 → ∃*𝑦(𝑦𝐵𝜑)))
54bicomi 227 . . . . 5 ((𝑥𝐴 → ∃*𝑦(𝑦𝐵𝜑)) ↔ ∃*𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)))
6 3anass 1096 . . . . . . 7 ((𝑥𝐴𝑦𝐵𝜑) ↔ (𝑥𝐴 ∧ (𝑦𝐵𝜑)))
76bicomi 227 . . . . . 6 ((𝑥𝐴 ∧ (𝑦𝐵𝜑)) ↔ (𝑥𝐴𝑦𝐵𝜑))
87mobii 2549 . . . . 5 (∃*𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)) ↔ ∃*𝑦(𝑥𝐴𝑦𝐵𝜑))
95, 8bitri 278 . . . 4 ((𝑥𝐴 → ∃*𝑦(𝑦𝐵𝜑)) ↔ ∃*𝑦(𝑥𝐴𝑦𝐵𝜑))
109albii 1826 . . 3 (∀𝑥(𝑥𝐴 → ∃*𝑦(𝑦𝐵𝜑)) ↔ ∀𝑥∃*𝑦(𝑥𝐴𝑦𝐵𝜑))
113, 10bitri 278 . 2 (∀𝑥𝐴 ∃*𝑦(𝑦𝐵𝜑) ↔ ∀𝑥∃*𝑦(𝑥𝐴𝑦𝐵𝜑))
122, 11bitri 278 1 (∀𝑥𝐴 ∃*𝑦𝐵 𝜑 ↔ ∀𝑥∃*𝑦(𝑥𝐴𝑦𝐵𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1088  wal 1540  wcel 2114  ∃*wmo 2539  wral 3054  ∃*wrmo 3057
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975
This theorem depends on definitions:  df-bi 210  df-an 400  df-3an 1090  df-ex 1787  df-mo 2541  df-ral 3059  df-rmo 3062
This theorem is referenced by:  2reu5lem3  3661
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