Theorem bj-sseq 13337

Description: If two converse inclusions are characterized each by a formula, then equality is characterized by the conjunction of these formulas. (Contributed by BJ, 30-Nov-2019.)

Hypotheses

Ref	Expression
bj-sseq.1	|- ( ph -> ( ps <-> A C_ B ) )
bj-sseq.2	|- ( ph -> ( ch <-> B C_ A ) )

Assertion

Ref	Expression
bj-sseq	|- ( ph -> ( ( ps /\ ch ) <-> A = B ) )

Proof of Theorem bj-sseq

Step	Hyp	Ref	Expression
1		bj-sseq.1	. . 3 |- ( ph -> ( ps <-> A C_ B ) )
2		bj-sseq.2	. . 3 |- ( ph -> ( ch <-> B C_ A ) )
3	1, 2	anbi12d 465	. 2 |- ( ph -> ( ( ps /\ ch ) <-> ( A C_ B /\ B C_ A ) ) )
4		eqss 3143	. 2 |- ( A = B <-> ( A C_ B /\ B C_ A ) )
5	3, 4	bitr4di 197	1 |- ( ph -> ( ( ps /\ ch ) <-> A = B ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1335   C_ wss 3102

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-11 1486  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-in 3108  df-ss 3115

This theorem is referenced by: (None)