Theorem ssdifsym 3863

Description: Symmetric class differences for subclasses. (Contributed by AV, 3-Jan-2022.)

Assertion

Ref	Expression
ssdifsym	|- ( ( A C_ V /\ B C_ V ) -> ( B = ( V \ A ) <-> A = ( V \ B ) ) )

Proof of Theorem ssdifsym

Step	Hyp	Ref	Expression
1		ssdifim 3862	. . . 4 |- ( ( A C_ V /\ B = ( V \ A ) ) -> A = ( V \ B ) )
2	1	ex 450	. . 3 |- ( A C_ V -> ( B = ( V \ A ) -> A = ( V \ B ) ) )
3	2	adantr 481	. 2 |- ( ( A C_ V /\ B C_ V ) -> ( B = ( V \ A ) -> A = ( V \ B ) ) )
4		ssdifim 3862	. . . 4 |- ( ( B C_ V /\ A = ( V \ B ) ) -> B = ( V \ A ) )
5	4	ex 450	. . 3 |- ( B C_ V -> ( A = ( V \ B ) -> B = ( V \ A ) ) )
6	5	adantl 482	. 2 |- ( ( A C_ V /\ B C_ V ) -> ( A = ( V \ B ) -> B = ( V \ A ) ) )
7	3, 6	impbid 202	1 |- ( ( A C_ V /\ B C_ V ) -> ( B = ( V \ A ) <-> A = ( V \ B ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   \ cdif 3571   C_ wss 3574

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-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rab 2921  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588

This theorem is referenced by: (None)