Theorem sbcalg 3051

Description: Move universal quantifier in and out of class substitution. (Contributed by NM, 16-Jan-2004.)

Assertion

Ref	Expression
sbcalg	|- ( A e. V -> ( [. A / y ]. A. x ph <-> A. x [. A / y ]. ph ) )

Distinct variable groups:   x, A   x, y

Allowed substitution hints:   ph( x, y)   A( y)   V( x, y)

Proof of Theorem sbcalg

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfsbcq2 3001	. 2 |- ( z = A -> ( [ z / y ] A. x ph <-> [. A / y ]. A. x ph ) )
2		dfsbcq2 3001	. . 3 |- ( z = A -> ( [ z / y ] ph <-> [. A / y ]. ph ) )
3	2	albidv 1847	. 2 |- ( z = A -> ( A. x [ z / y ] ph <-> A. x [. A / y ]. ph ) )
4		sbal 2028	. 2 |- ( [ z / y ] A. x ph <-> A. x [ z / y ] ph )
5	1, 3, 4	vtoclbg 2834	1 |- ( A e. V -> ( [. A / y ]. A. x ph <-> A. x [. A / y ]. ph ) )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1371   = wceq 1373   [wsb 1785   e. wcel 2176   [.wsbc 2998

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-sbc 2999

This theorem is referenced by:  sbcabel  3080  sbcssg  3569