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Metamath Proof Explorer

Theorem 2ax6elem

Description: We can always find values matching x and y , as long as they are represented by distinct variables. This theorem merges two ax6e instances E. z z = x and E. w w = y into a common expression. Alan Sare contributed a variant of this theorem with distinct variable conditions before, see ax6e2nd . Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by Wolf Lammen, 27-Sep-2018) (New usage is discouraged.)

Ref Expression
Assertion 2ax6elem 
|- ( -. A. w w = z -> E. z E. w ( z = x /\ w = y ) )

Proof

Step Hyp Ref Expression
1 ax6e 
 |-  E. z z = x
2 nfnae 
 |-  F/ z -. A. w w = z
3 nfnae 
 |-  F/ z -. A. w w = x
4 2 3 nfan 
 |-  F/ z ( -. A. w w = z /\ -. A. w w = x )
5 nfeqf 
 |-  ( ( -. A. w w = z /\ -. A. w w = x ) -> F/ w z = x )
6 pm3.21 
 |-  ( w = y -> ( z = x -> ( z = x /\ w = y ) ) )
7 5 6 spimed 
 |-  ( ( -. A. w w = z /\ -. A. w w = x ) -> ( z = x -> E. w ( z = x /\ w = y ) ) )
8 4 7 eximd 
 |-  ( ( -. A. w w = z /\ -. A. w w = x ) -> ( E. z z = x -> E. z E. w ( z = x /\ w = y ) ) )
9 1 8 mpi 
 |-  ( ( -. A. w w = z /\ -. A. w w = x ) -> E. z E. w ( z = x /\ w = y ) )
10 9 ex 
 |-  ( -. A. w w = z -> ( -. A. w w = x -> E. z E. w ( z = x /\ w = y ) ) )
11 ax6e 
 |-  E. z z = y
12 nfae 
 |-  F/ z A. w w = x
13 equvini 
 |-  ( z = y -> E. w ( z = w /\ w = y ) )
14 equtrr 
 |-  ( w = x -> ( z = w -> z = x ) )
15 14 anim1d 
 |-  ( w = x -> ( ( z = w /\ w = y ) -> ( z = x /\ w = y ) ) )
16 15 aleximi 
 |-  ( A. w w = x -> ( E. w ( z = w /\ w = y ) -> E. w ( z = x /\ w = y ) ) )
17 13 16 syl5 
 |-  ( A. w w = x -> ( z = y -> E. w ( z = x /\ w = y ) ) )
18 12 17 eximd 
 |-  ( A. w w = x -> ( E. z z = y -> E. z E. w ( z = x /\ w = y ) ) )
19 11 18 mpi 
 |-  ( A. w w = x -> E. z E. w ( z = x /\ w = y ) )
20 10 19 pm2.61d2 
 |-  ( -. A. w w = z -> E. z E. w ( z = x /\ w = y ) )