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Metamath Proof Explorer

Theorem dral1-o

Description: Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of Megill p. 448 (p. 16 of preprint). Version of dral1 using ax-c11 . (Contributed by NM, 24-Nov-1994) (New usage is discouraged.)

Ref Expression
Hypothesis dral1-o.1 
|- ( A. x x = y -> ( ph <-> ps ) )
Assertion dral1-o 
|- ( A. x x = y -> ( A. x ph <-> A. y ps ) )

Proof

Step Hyp Ref Expression
1 dral1-o.1 
 |-  ( A. x x = y -> ( ph <-> ps ) )
2 hbae-o 
 |-  ( A. x x = y -> A. x A. x x = y )
3 1 biimpd 
 |-  ( A. x x = y -> ( ph -> ps ) )
4 2 3 alimdh 
 |-  ( A. x x = y -> ( A. x ph -> A. x ps ) )
5 ax-c11 
 |-  ( A. x x = y -> ( A. x ps -> A. y ps ) )
6 4 5 syld 
 |-  ( A. x x = y -> ( A. x ph -> A. y ps ) )
7 hbae-o 
 |-  ( A. x x = y -> A. y A. x x = y )
8 1 biimprd 
 |-  ( A. x x = y -> ( ps -> ph ) )
9 7 8 alimdh 
 |-  ( A. x x = y -> ( A. y ps -> A. y ph ) )
10 ax-c11 
 |-  ( A. y y = x -> ( A. y ph -> A. x ph ) )
11 10 aecoms-o 
 |-  ( A. x x = y -> ( A. y ph -> A. x ph ) )
12 9 11 syld 
 |-  ( A. x x = y -> ( A. y ps -> A. x ph ) )
13 6 12 impbid 
 |-  ( A. x x = y -> ( A. x ph <-> A. y ps ) )