Theorem sbceq1ddi 33928

Description: A lemma for eliminating inequality, in inference form. (Contributed by Giovanni Mascellani, 31-May-2019.)

Hypotheses

Ref	Expression
sbceq1ddi.1	|- ( ph -> A = B )
sbceq1ddi.2	|- ( ps -> th )
sbceq1ddi.3	|- ( [. A / x ]. ch <-> th )
sbceq1ddi.4	|- ( [. B / x ]. ch <-> et )

Assertion

Ref	Expression
sbceq1ddi	|- ( ( ph /\ ps ) -> et )

Proof of Theorem sbceq1ddi

Step	Hyp	Ref	Expression
1		sbceq1ddi.1	. . . 4 |- ( ph -> A = B )
2	1	adantr 481	. . 3 |- ( ( ph /\ ps ) -> A = B )
3		sbceq1ddi.2	. . . . 5 |- ( ps -> th )
4		sbceq1ddi.3	. . . . 5 |- ( [. A / x ]. ch <-> th )
5	3, 4	sylibr 224	. . . 4 |- ( ps -> [. A / x ]. ch )
6	5	adantl 482	. . 3 |- ( ( ph /\ ps ) -> [. A / x ]. ch )
7	2, 6	sbceq1dd 3441	. 2 |- ( ( ph /\ ps ) -> [. B / x ]. ch )
8		sbceq1ddi.4	. 2 |- ( [. B / x ]. ch <-> et )
9	7, 8	sylib 208	1 |- ( ( ph /\ ps ) -> et )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   [.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-ext 2602

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-cleq 2615  df-clel 2618  df-sbc 3436

This theorem is referenced by: (None)