Theorem sbcbi1 3483

Description: Distribution of class substitution over biconditional. One direction of sbcbig 3480 that holds for proper classes. (Contributed by NM, 17-Aug-2018.)

Assertion

Ref	Expression
sbcbi1	|- ( [. A / x ]. ( ph <-> ps ) -> ( [. A / x ]. ph <-> [. A / x ]. ps ) )

Proof of Theorem sbcbi1

Step	Hyp	Ref	Expression
1		sbcex 3445	. 2 |- ( [. A / x ]. ( ph <-> ps ) -> A e. _V )
2		sbcbig 3480	. . 3 |- ( A e. _V -> ( [. A / x ]. ( ph <-> ps ) <-> ( [. A / x ]. ph <-> [. A / x ]. ps ) ) )
3	2	biimpd 219	. 2 |- ( A e. _V -> ( [. A / x ]. ( ph <-> ps ) -> ( [. A / x ]. ph <-> [. A / x ]. ps ) ) )
4	1, 3	mpcom 38	1 |- ( [. A / x ]. ( ph <-> ps ) -> ( [. A / x ]. ph <-> [. A / x ]. ps ) )

Syntax hints:   -> wi 4   <-> wb 196   e. wcel 1990   _Vcvv 3200   [.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:  dfconngr1  27048