Theorem eupth2lem1 27078

Description: Lemma for eupth2 27099. (Contributed by Mario Carneiro, 8-Apr-2015.)

Assertion

Ref	Expression
eupth2lem1	|- ( U e. V -> ( U e. if ( A = B , (/) , { A , B } ) <-> ( A =/= B /\ ( U = A \/ U = B ) ) ) )

Proof of Theorem eupth2lem1

Step	Hyp	Ref	Expression
1		eleq2 2690	. . 3 |- ( (/) = if ( A = B , (/) , { A , B } ) -> ( U e. (/) <-> U e. if ( A = B , (/) , { A , B } ) ) )
2	1	bibi1d 333	. 2 |- ( (/) = if ( A = B , (/) , { A , B } ) -> ( ( U e. (/) <-> ( A =/= B /\ ( U = A \/ U = B ) ) ) <-> ( U e. if ( A = B , (/) , { A , B } ) <-> ( A =/= B /\ ( U = A \/ U = B ) ) ) ) )
3		eleq2 2690	. . 3 |- ( { A , B } = if ( A = B , (/) , { A , B } ) -> ( U e. { A , B } <-> U e. if ( A = B , (/) , { A , B } ) ) )
4	3	bibi1d 333	. 2 |- ( { A , B } = if ( A = B , (/) , { A , B } ) -> ( ( U e. { A , B } <-> ( A =/= B /\ ( U = A \/ U = B ) ) ) <-> ( U e. if ( A = B , (/) , { A , B } ) <-> ( A =/= B /\ ( U = A \/ U = B ) ) ) ) )
5		noel 3919	. . . 4 |- -. U e. (/)
6	5	a1i 11	. . 3 |- ( ( U e. V /\ A = B ) -> -. U e. (/) )
7		simpl 473	. . . . 5 |- ( ( A =/= B /\ ( U = A \/ U = B ) ) -> A =/= B )
8	7	neneqd 2799	. . . 4 |- ( ( A =/= B /\ ( U = A \/ U = B ) ) -> -. A = B )
9		simpr 477	. . . 4 |- ( ( U e. V /\ A = B ) -> A = B )
10	8, 9	nsyl3 133	. . 3 |- ( ( U e. V /\ A = B ) -> -. ( A =/= B /\ ( U = A \/ U = B ) ) )
11	6, 10	2falsed 366	. 2 |- ( ( U e. V /\ A = B ) -> ( U e. (/) <-> ( A =/= B /\ ( U = A \/ U = B ) ) ) )
12		elprg 4196	. . 3 |- ( U e. V -> ( U e. { A , B } <-> ( U = A \/ U = B ) ) )
13		df-ne 2795	. . . 4 |- ( A =/= B <-> -. A = B )
14		ibar 525	. . . 4 |- ( A =/= B -> ( ( U = A \/ U = B ) <-> ( A =/= B /\ ( U = A \/ U = B ) ) ) )
15	13, 14	sylbir 225	. . 3 |- ( -. A = B -> ( ( U = A \/ U = B ) <-> ( A =/= B /\ ( U = A \/ U = B ) ) ) )
16	12, 15	sylan9bb 736	. 2 |- ( ( U e. V /\ -. A = B ) -> ( U e. { A , B } <-> ( A =/= B /\ ( U = A \/ U = B ) ) ) )
17	2, 4, 11, 16	ifbothda 4123	1 |- ( U e. V -> ( U e. if ( A = B , (/) , { A , B } ) <-> ( A =/= B /\ ( U = A \/ U = B ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   (/)c0 3915   ifcif 4086   {cpr 4179

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-ne 2795  df-v 3202  df-dif 3577  df-un 3579  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180

This theorem is referenced by:  eupth2lem2  27079  eupth2lem3lem6  27093