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Theorem hbra2VD 31951
Description: Virtual deduction proof of nfra2 2889. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.
1::  |-  ( A. y  e.  B A. x  e.  A ph  ->  A. y A. y  e.  B A. x  e.  A ph )
2::  |-  ( A. x  e.  A A. y  e.  B ph  <->  A. y  e.  B A. x  e.  A ph )
3:1,2,?: e00 31856  |-  ( A. x  e.  A A. y  e.  B ph  ->  A. y A. y  e.  B A. x  e.  A ph )
4:2:  |-  A. y ( A. x  e.  A A. y  e.  B ph  <->  A. y  e.  B A. x  e.  A ph )
5:4,?: e0a 31860  |-  ( A. y A. x  e.  A A. y  e.  B ph  <->  A. y A. y  e.  B A. x  e.  A ph )
qed:3,5,?: e00 31856  |-  ( A. x  e.  A A. y  e.  B ph  ->  A. y A. x  e.  A A. y  e.  B ph )
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
hbra2VD  |-  ( A. x  e.  A  A. y  e.  B  ph  ->  A. y A. x  e.  A  A. y  e.  B  ph )
Distinct variable groups:    y, A    x, B    x, y
Allowed substitution hints:    ph( x, y)    A( x)    B( y)

Proof of Theorem hbra2VD
StepHypRef Expression
1 ralcom 2987 . 2  |-  ( A. x  e.  A  A. y  e.  B  ph  <->  A. y  e.  B  A. x  e.  A  ph )
2 hbra1 2811 . 2  |-  ( A. y  e.  B  A. x  e.  A  ph  ->  A. y A. y  e.  B  A. x  e.  A  ph )
31, 2hbxfrbi 1614 1  |-  ( A. x  e.  A  A. y  e.  B  ph  ->  A. y A. x  e.  A  A. y  e.  B  ph )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1368   A.wral 2799
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-ex 1588  df-nf 1591  df-sb 1703  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2804
This theorem is referenced by: (None)
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