Theorem bj-axc14nf 32838

Description: Proof of a version of axc14 2372 using the "non-free" idiom. (Contributed by BJ, 20-Oct-2021.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-axc14nf	|- ( -. A. z z = x -> ( -. A. z z = y -> F/ z x e. y ) )

Proof of Theorem bj-axc14nf

Dummy variable t is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfnae 2318	. 2 |- F/ z -. A. z z = x
2		bj-nfeel2 32837	. 2 |- ( -. A. z z = x -> F/ z x e. t )
3		elequ2 2004	. 2 |- ( t = y -> ( x e. t <-> x e. y ) )
4	1, 2, 3	bj-dvelimdv1 32835	1 |- ( -. A. z z = x -> ( -. A. z z = y -> F/ z x e. y ) )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1481   F/wnf 1708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710

This theorem is referenced by:  bj-axc14  32839