Metamath Proof Explorer

Theorem cgsex4gOLD

Description: Obsolete version of cgsex4g as of 28-Jun-2024. (Contributed by NM, 5-Aug-1995) Avoid ax-10 . (Revised by Gino Giotto, 20-Aug-2023) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Hypotheses cgsex4gOLD.1 
|- ( ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) ) -> ch )
cgsex4gOLD.2 
|- ( ch -> ( ph <-> ps ) )
Assertion cgsex4gOLD 
|- ( ( ( A e. R /\ B e. S ) /\ ( C e. R /\ D e. S ) ) -> ( E. x E. y E. z E. w ( ch /\ ph ) <-> ps ) )

Proof

Step Hyp Ref Expression
1 cgsex4gOLD.1 
 |-  ( ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) ) -> ch )
2 cgsex4gOLD.2 
 |-  ( ch -> ( ph <-> ps ) )
3 2 biimpa 
 |-  ( ( ch /\ ph ) -> ps )
4 3 exlimivv 
 |-  ( E. z E. w ( ch /\ ph ) -> ps )
5 4 exlimivv 
 |-  ( E. x E. y E. z E. w ( ch /\ ph ) -> ps )
6 elisset 
 |-  ( A e. R -> E. x x = A )
7 elisset 
 |-  ( B e. S -> E. y y = B )
8 6 7 anim12i 
 |-  ( ( A e. R /\ B e. S ) -> ( E. x x = A /\ E. y y = B ) )
9 exdistrv 
 |-  ( E. x E. y ( x = A /\ y = B ) <-> ( E. x x = A /\ E. y y = B ) )
10 8 9 sylibr 
 |-  ( ( A e. R /\ B e. S ) -> E. x E. y ( x = A /\ y = B ) )
11 elisset 
 |-  ( C e. R -> E. z z = C )
12 elisset 
 |-  ( D e. S -> E. w w = D )
13 11 12 anim12i 
 |-  ( ( C e. R /\ D e. S ) -> ( E. z z = C /\ E. w w = D ) )
14 exdistrv 
 |-  ( E. z E. w ( z = C /\ w = D ) <-> ( E. z z = C /\ E. w w = D ) )
15 13 14 sylibr 
 |-  ( ( C e. R /\ D e. S ) -> E. z E. w ( z = C /\ w = D ) )
16 10 15 anim12i 
 |-  ( ( ( A e. R /\ B e. S ) /\ ( C e. R /\ D e. S ) ) -> ( E. x E. y ( x = A /\ y = B ) /\ E. z E. w ( z = C /\ w = D ) ) )
17 excom 
 |-  ( E. y E. z E. w ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) ) <-> E. z E. y E. w ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) ) )
18 17 exbii 
 |-  ( E. x E. y E. z E. w ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) ) <-> E. x E. z E. y E. w ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) ) )
19 4exdistrv 
 |-  ( E. x E. z E. y E. w ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) ) <-> ( E. x E. y ( x = A /\ y = B ) /\ E. z E. w ( z = C /\ w = D ) ) )
20 18 19 bitri 
 |-  ( E. x E. y E. z E. w ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) ) <-> ( E. x E. y ( x = A /\ y = B ) /\ E. z E. w ( z = C /\ w = D ) ) )
21 16 20 sylibr 
 |-  ( ( ( A e. R /\ B e. S ) /\ ( C e. R /\ D e. S ) ) -> E. x E. y E. z E. w ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) ) )
22 1 2eximi 
 |-  ( E. z E. w ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) ) -> E. z E. w ch )
23 22 2eximi 
 |-  ( E. x E. y E. z E. w ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) ) -> E. x E. y E. z E. w ch )
24 21 23 syl 
 |-  ( ( ( A e. R /\ B e. S ) /\ ( C e. R /\ D e. S ) ) -> E. x E. y E. z E. w ch )
25 2 biimprcd 
 |-  ( ps -> ( ch -> ph ) )
26 25 ancld 
 |-  ( ps -> ( ch -> ( ch /\ ph ) ) )
27 26 2eximdv 
 |-  ( ps -> ( E. z E. w ch -> E. z E. w ( ch /\ ph ) ) )
28 27 2eximdv 
 |-  ( ps -> ( E. x E. y E. z E. w ch -> E. x E. y E. z E. w ( ch /\ ph ) ) )
29 24 28 syl5com 
 |-  ( ( ( A e. R /\ B e. S ) /\ ( C e. R /\ D e. S ) ) -> ( ps -> E. x E. y E. z E. w ( ch /\ ph ) ) )
30 5 29 impbid2 
 |-  ( ( ( A e. R /\ B e. S ) /\ ( C e. R /\ D e. S ) ) -> ( E. x E. y E. z E. w ( ch /\ ph ) <-> ps ) )