Theorem bj-sseq 9931

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 442	. 2 |- ( ph -> ( ( ps /\ ch ) <-> ( A C_ B /\ B C_ A ) ) )
4		eqss 2960	. 2 |- ( A = B <-> ( A C_ B /\ B C_ A ) )
5	3, 4	syl6bbr 187	1 |- ( ph -> ( ( ps /\ ch ) <-> A = B ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1243   C_ wss 2917

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-in 2924  df-ss 2931

This theorem is referenced by: (None)