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Theorem r2exf 2650
 Description: Double restricted existential quantification. (Contributed by Mario Carneiro, 14-Oct-2016.)
Hypothesis
Ref Expression
r2alf.1 yA
Assertion
Ref Expression
r2exf (x A y B φxy((x A y B) φ))
Distinct variable group:   x,y
Allowed substitution hints:   φ(x,y)   A(x,y)   B(x,y)

Proof of Theorem r2exf
StepHypRef Expression
1 df-rex 2620 . 2 (x A y B φx(x A y B φ))
2 r2alf.1 . . . . . 6 yA
32nfcri 2483 . . . . 5 y x A
4319.42 1880 . . . 4 (y(x A (y B φ)) ↔ (x A y(y B φ)))
5 anass 630 . . . . 5 (((x A y B) φ) ↔ (x A (y B φ)))
65exbii 1582 . . . 4 (y((x A y B) φ) ↔ y(x A (y B φ)))
7 df-rex 2620 . . . . 5 (y B φy(y B φ))
87anbi2i 675 . . . 4 ((x A y B φ) ↔ (x A y(y B φ)))
94, 6, 83bitr4i 268 . . 3 (y((x A y B) φ) ↔ (x A y B φ))
109exbii 1582 . 2 (xy((x A y B) φ) ↔ x(x A y B φ))
111, 10bitr4i 243 1 (x A y B φxy((x A y B) φ))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541   ∈ wcel 1710  Ⅎwnfc 2476  ∃wrex 2615 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  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-cleq 2346  df-clel 2349  df-nfc 2478  df-rex 2620 This theorem is referenced by:  r2ex  2652  rexcomf  2770
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