Theorem elpreq 29360

Description: Equality wihin a pair. (Contributed by Thierry Arnoux, 23-Aug-2017.)

Hypotheses

Ref	Expression
elpreq.1	|- ( ph -> X e. { A , B } )
elpreq.2	|- ( ph -> Y e. { A , B } )
elpreq.3	|- ( ph -> ( X = A <-> Y = A ) )

Assertion

Ref	Expression
elpreq	|- ( ph -> X = Y )

Proof of Theorem elpreq

Step	Hyp	Ref	Expression
1		simpr 477	. . 3 |- ( ( ph /\ X = A ) -> X = A )
2		elpreq.3	. . . 4 |- ( ph -> ( X = A <-> Y = A ) )
3	2	biimpa 501	. . 3 |- ( ( ph /\ X = A ) -> Y = A )
4	1, 3	eqtr4d 2659	. 2 |- ( ( ph /\ X = A ) -> X = Y )
5		elpreq.1	. . . . 5 |- ( ph -> X e. { A , B } )
6		elpri 4197	. . . . 5 |- ( X e. { A , B } -> ( X = A \/ X = B ) )
7	5, 6	syl 17	. . . 4 |- ( ph -> ( X = A \/ X = B ) )
8	7	orcanai 952	. . 3 |- ( ( ph /\ -. X = A ) -> X = B )
9		simpl 473	. . . 4 |- ( ( ph /\ -. X = A ) -> ph )
10	2	notbid 308	. . . . 5 |- ( ph -> ( -. X = A <-> -. Y = A ) )
11	10	biimpa 501	. . . 4 |- ( ( ph /\ -. X = A ) -> -. Y = A )
12		elpreq.2	. . . . 5 |- ( ph -> Y e. { A , B } )
13		elpri 4197	. . . . 5 |- ( Y e. { A , B } -> ( Y = A \/ Y = B ) )
14		pm2.53 388	. . . . 5 |- ( ( Y = A \/ Y = B ) -> ( -. Y = A -> Y = B ) )
15	12, 13, 14	3syl 18	. . . 4 |- ( ph -> ( -. Y = A -> Y = B ) )
16	9, 11, 15	sylc 65	. . 3 |- ( ( ph /\ -. X = A ) -> Y = B )
17	8, 16	eqtr4d 2659	. 2 |- ( ( ph /\ -. X = A ) -> X = Y )
18	4, 17	pm2.61dan 832	1 |- ( ph -> X = Y )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   {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-v 3202  df-un 3579  df-sn 4178  df-pr 4180

This theorem is referenced by:  indpreima  30087