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Metamath Proof Explorer

Theorem eqelbid

Description: A variable elimination law for equality within a given set A . See equvel . (Contributed by Thierry Arnoux, 20-Feb-2025)

Ref Expression
Hypotheses eqelbid.1 
|- ( ph -> B e. A )
eqelbid.2 
|- ( ph -> C e. A )
Assertion eqelbid 
|- ( ph -> ( A. x e. A ( x = B <-> x = C ) <-> B = C ) )

Proof

Step Hyp Ref Expression
1 eqelbid.1 
 |-  ( ph -> B e. A )
2 eqelbid.2 
 |-  ( ph -> C e. A )
3 eqeq1 
 |-  ( x = B -> ( x = B <-> B = B ) )
4 eqeq1 
 |-  ( x = B -> ( x = C <-> B = C ) )
5 3 4 bibi12d 
 |-  ( x = B -> ( ( x = B <-> x = C ) <-> ( B = B <-> B = C ) ) )
6 eqid 
 |-  B = B
7 6 tbt 
 |-  ( B = C <-> ( B = C <-> B = B ) )
8 bicom 
 |-  ( ( B = C <-> B = B ) <-> ( B = B <-> B = C ) )
9 7 8 bitri 
 |-  ( B = C <-> ( B = B <-> B = C ) )
10 5 9 bitr4di 
 |-  ( x = B -> ( ( x = B <-> x = C ) <-> B = C ) )
11 simpr 
 |-  ( ( ph /\ A. x e. A ( x = B <-> x = C ) ) -> A. x e. A ( x = B <-> x = C ) )
12 1 adantr 
 |-  ( ( ph /\ A. x e. A ( x = B <-> x = C ) ) -> B e. A )
13 10 11 12 rspcdva 
 |-  ( ( ph /\ A. x e. A ( x = B <-> x = C ) ) -> B = C )
14 simplr 
 |-  ( ( ( ph /\ B = C ) /\ x e. A ) -> B = C )
15 14 eqeq2d 
 |-  ( ( ( ph /\ B = C ) /\ x e. A ) -> ( x = B <-> x = C ) )
16 15 ralrimiva 
 |-  ( ( ph /\ B = C ) -> A. x e. A ( x = B <-> x = C ) )
17 13 16 impbida 
 |-  ( ph -> ( A. x e. A ( x = B <-> x = C ) <-> B = C ) )