Theorem bj-sseq 13673

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 3157	. 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 1343   C_ wss 3116

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-in 3122  df-ss 3129

This theorem is referenced by: (None)