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Metamath Proof Explorer

Theorem eqeq1dALT

Description: Alternate proof of eqeq1d , shorter but requiring ax-12 . (Contributed by NM, 27-Dec-1993) (Revised by Wolf Lammen, 19-Nov-2019) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Hypothesis eqeq1d.1 
|- ( ph -> A = B )
Assertion eqeq1dALT 
|- ( ph -> ( A = C <-> B = C ) )

Proof

Step Hyp Ref Expression
1 eqeq1d.1 
 |-  ( ph -> A = B )
2 dfcleq 
 |-  ( A = B <-> A. x ( x e. A <-> x e. B ) )
3 1 2 sylib 
 |-  ( ph -> A. x ( x e. A <-> x e. B ) )
4 3 19.21bi 
 |-  ( ph -> ( x e. A <-> x e. B ) )
5 4 bibi1d 
 |-  ( ph -> ( ( x e. A <-> x e. C ) <-> ( x e. B <-> x e. C ) ) )
6 5 albidv 
 |-  ( ph -> ( A. x ( x e. A <-> x e. C ) <-> A. x ( x e. B <-> x e. C ) ) )
7 dfcleq 
 |-  ( A = C <-> A. x ( x e. A <-> x e. C ) )
8 dfcleq 
 |-  ( B = C <-> A. x ( x e. B <-> x e. C ) )
9 6 7 8 3bitr4g 
 |-  ( ph -> ( A = C <-> B = C ) )