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Theorem ralxpxfr2d 26695
 Description: Transfer a universal quantifier between one variable with pair-like semantics and two. (Contributed by Stefan O'Rear, 27-Feb-2015.)
Hypotheses
Ref Expression
ralxpxfr2d.a
ralxpxfr2d.b
ralxpxfr2d.c
Assertion
Ref Expression
ralxpxfr2d
Distinct variable groups:   ,,   ,,   ,   ,   ,   ,   ,   ,
Allowed substitution hints:   ()   (,)   (,)   (,,)   (,)   (,)

Proof of Theorem ralxpxfr2d
StepHypRef Expression
1 df-ral 2702 . . . 4
2 ralxpxfr2d.b . . . . . 6
32imbi1d 309 . . . . 5
43albidv 1635 . . . 4
51, 4syl5bb 249 . . 3
6 ralcom4 2966 . . . 4
7 ralcom4 2966 . . . . 5
87ralbii 2721 . . . 4
9 r19.23v 2814 . . . . . . 7
109ralbii 2721 . . . . . 6
11 r19.23v 2814 . . . . . 6
1210, 11bitr2i 242 . . . . 5
1312albii 1575 . . . 4
146, 8, 133bitr4ri 270 . . 3
155, 14syl6bb 253 . 2
16 ralxpxfr2d.c . . . . . 6
1716pm5.74da 669 . . . . 5
1817albidv 1635 . . . 4
19 ralxpxfr2d.a . . . . 5
20 biidd 229 . . . . 5
2119, 20ceqsalv 2974 . . . 4
2218, 21syl6bb 253 . . 3
23222ralbidv 2739 . 2
2415, 23bitrd 245 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359  wal 1549   wceq 1652   wcel 1725  wral 2697  wrex 2698  cvv 2948 This theorem is referenced by:  ralxpmap  26696 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-v 2950
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