Theorem sbcalg 3094

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.xph <-> 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 3049	. 2 |- (z = A -> ([z / y]A.xph <-> [.A / y ].A.xph))
2		dfsbcq2 3049	. . 3 |- (z = A -> ([z / y]ph <-> [.A / y ].ph))
3	2	albidv 1625	. 2 |- (z = A -> (A.x[z / y]ph <-> A.x [.A / y ].ph))
4		sbal 2127	. 2 |- ([z / y]A.xph <-> A.x[z / y]ph)
5	1, 3, 4	vtoclbg 2915	1 |- (A e. V -> ( [.A / y ].A.xph <-> A.x [.A / y ].ph))

Syntax hints:   -> wi 4   <-> wb 176  A.wal 1540   = wceq 1642  [wsb 1648   e. wcel 1710   [.wsbc 3046

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-sbc 3047

This theorem is referenced by:  sbcabel  3123  sbcss  3660