Theorem bj-axc14 32839

Description: Alternate proof of axc14 2372 (even when inlining the above results, this gives a shorter proof). (Contributed by BJ, 20-Oct-2021.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-axc14	|- ( -. A. z z = x -> ( -. A. z z = y -> ( x e. y -> A. z x e. y ) ) )

Proof of Theorem bj-axc14

Step	Hyp	Ref	Expression
1		bj-axc14nf 32838	. 2 |- ( -. A. z z = x -> ( -. A. z z = y -> F/ z x e. y ) )
2		nf5r 2064	. . 3 |- ( F/ z x e. y -> ( x e. y -> A. z x e. y ) )
3	2	a1i 11	. 2 |- ( -. A. z z = x -> ( F/ z x e. y -> ( x e. y -> A. z x e. y ) ) )
4	1, 3	syld 47	1 |- ( -. A. z z = x -> ( -. A. z z = y -> ( x e. y -> A. 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: (None)