Theorem bj-sseq 14583

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 473	. 2 |- ( ph -> ( ( ps /\ ch ) <-> ( A C_ B /\ B C_ A ) ) )
4		eqss 3172	. 2 |- ( A = B <-> ( A C_ B /\ B C_ A ) )
5	3, 4	bitr4di 198	1 |- ( ph -> ( ( ps /\ ch ) <-> A = B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   C_ wss 3131

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-in 3137  df-ss 3144

This theorem is referenced by: (None)