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Theorem 2reu5lem2 3042
 Description: Lemma for 2reu5 3044. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
Assertion
Ref Expression
2reu5lem2 (x A ∃*y B φx∃*y(x A y B φ))
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 2622 . . 3 (∃*y B φ∃*y(y B φ))
21ralbii 2638 . 2 (x A ∃*y B φx A ∃*y(y B φ))
3 df-ral 2619 . . 3 (x A ∃*y(y B φ) ↔ x(x A∃*y(y B φ)))
4 moanimv 2262 . . . . . 6 (∃*y(x A (y B φ)) ↔ (x A∃*y(y B φ)))
54bicomi 193 . . . . 5 ((x A∃*y(y B φ)) ↔ ∃*y(x A (y B φ)))
6 3anass 938 . . . . . . 7 ((x A y B φ) ↔ (x A (y B φ)))
76bicomi 193 . . . . . 6 ((x A (y B φ)) ↔ (x A y B φ))
87mobii 2240 . . . . 5 (∃*y(x A (y B φ)) ↔ ∃*y(x A y B φ))
95, 8bitri 240 . . . 4 ((x A∃*y(y B φ)) ↔ ∃*y(x A y B φ))
109albii 1566 . . 3 (x(x A∃*y(y B φ)) ↔ x∃*y(x A y B φ))
113, 10bitri 240 . 2 (x A ∃*y(y B φ) ↔ x∃*y(x A y B φ))
122, 11bitri 240 1 (x A ∃*y B φx∃*y(x A y B φ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∀wal 1540   ∈ wcel 1710  ∃*wmo 2205  ∀wral 2614  ∃*wrmo 2617 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-ral 2619  df-rmo 2622 This theorem is referenced by:  2reu5lem3  3043
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