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Theorem r2exf 2316
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 2286 . 2 (x A y B φx(x A y B φ))
2 r2alf.1 . . . . . 6 yA
32nfcri 2150 . . . . 5 y x A
4319.42 1556 . . . 4 (y(x A (y B φ)) ↔ (x A y(y B φ)))
5 anass 383 . . . . 5 (((x A y B) φ) ↔ (x A (y B φ)))
65exbii 1474 . . . 4 (y((x A y B) φ) ↔ y(x A (y B φ)))
7 df-rex 2286 . . . . 5 (y B φy(y B φ))
87anbi2i 433 . . . 4 ((x A y B φ) ↔ (x A y(y B φ)))
94, 6, 83bitr4i 201 . . 3 (y((x A y B) φ) ↔ (x A y B φ))
109exbii 1474 . 2 (xy((x A y B) φ) ↔ x(x A y B φ))
111, 10bitr4i 176 1 (x A y B φxy((x A y B) φ))
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98  wex 1358   wcel 1370  wnfc 2143  wrex 2281
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000
This theorem depends on definitions:  df-bi 110  df-nf 1326  df-sb 1624  df-cleq 2011  df-clel 2014  df-nfc 2145  df-rex 2286
This theorem is referenced by:  r2ex  2318  rexcomf  2446
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