Theorem sbcimi 33912

Description: Distribution of class substitution over implication, in inference form. (Contributed by Giovanni Mascellani, 27-May-2019.)

Hypotheses

Ref	Expression
sbcimi.1	|- A e. _V
sbcimi.2	|- ( [. A / x ]. ph <-> ch )
sbcimi.3	|- ( [. A / x ]. ps <-> et )

Assertion

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

Proof of Theorem sbcimi

Step	Hyp	Ref	Expression
1		sbcimi.1	. . 3 |- A e. _V
2		sbcimg 3477	. . 3 |- ( A e. _V -> ( [. A / x ]. ( ph -> ps ) <-> ( [. A / x ]. ph -> [. A / x ]. ps ) ) )
3	1, 2	ax-mp 5	. 2 |- ( [. A / x ]. ( ph -> ps ) <-> ( [. A / x ]. ph -> [. A / x ]. ps ) )
4		sbcimi.2	. . 3 |- ( [. A / x ]. ph <-> ch )
5		sbcimi.3	. . 3 |- ( [. A / x ]. ps <-> et )
6	4, 5	imbi12i 340	. 2 |- ( ( [. A / x ]. ph -> [. A / x ]. ps ) <-> ( ch -> et ) )
7	3, 6	bitri 264	1 |- ( [. A / x ]. ( ph -> ps ) <-> ( ch -> et ) )

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: (None)