Theorem sbcbi 38749

Description: Implication form of sbcbiiOLD 38741. sbcbi 38749 is sbcbiVD 39112 without virtual deductions and was automatically derived from sbcbiVD 39112 using the tools program translate..without..overwriting.cmd and Metamath's minimize command. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem sbcbi

Step	Hyp	Ref	Expression
1		spsbc 3448	. 2 |- ( A e. V -> ( A. x ( ph <-> ps ) -> [. A / x ]. ( ph <-> ps ) ) )
2		sbcbig 3480	. 2 |- ( A e. V -> ( [. A / x ]. ( ph <-> ps ) <-> ( [. A / x ]. ph <-> [. A / x ]. ps ) ) )
3	1, 2	sylibd 229	1 |- ( A e. V -> ( A. x ( ph <-> ps ) -> ( [. A / x ]. ph <-> [. A / x ]. ps ) ) )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   e. wcel 1990   [.wsbc 3435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-v 3202  df-sbc 3436

This theorem is referenced by:  trsbcVD  39113  sbcssgVD  39119  onfrALTlem5VD  39121