Theorem wl-cbvalnae 33320

Description: A more general version of cbval 2271 when non-free properties depend on a distinctor. Such expressions arise in proofs aiming at the elimination of distinct variable constraints, specifically in application of dvelimf 2334, nfsb2 2360 or dveeq1 2300. (Contributed by Wolf Lammen, 4-Jun-2019.)

Hypotheses

Ref	Expression
wl-cbvalnae.1	|- ( -. A. x x = y -> F/ y ph )
wl-cbvalnae.2	|- ( -. A. x x = y -> F/ x ps )
wl-cbvalnae.3	|- ( x = y -> ( ph <-> ps ) )

Assertion

Ref	Expression
wl-cbvalnae	|- ( A. x ph <-> A. y ps )

Proof of Theorem wl-cbvalnae

Step	Hyp	Ref	Expression
1		nftru 1730	. . 3 |- F/ x T.
2		nftru 1730	. . 3 |- F/ y T.
3		wl-cbvalnae.1	. . . 4 |- ( -. A. x x = y -> F/ y ph )
4	3	a1i 11	. . 3 |- ( T. -> ( -. A. x x = y -> F/ y ph ) )
5		wl-cbvalnae.2	. . . 4 |- ( -. A. x x = y -> F/ x ps )
6	5	a1i 11	. . 3 |- ( T. -> ( -. A. x x = y -> F/ x ps ) )
7		wl-cbvalnae.3	. . . 4 |- ( x = y -> ( ph <-> ps ) )
8	7	a1i 11	. . 3 |- ( T. -> ( x = y -> ( ph <-> ps ) ) )
9	1, 2, 4, 6, 8	wl-cbvalnaed 33319	. 2 |- ( T. -> ( A. x ph <-> A. y ps ) )
10	9	trud 1493	1 |- ( A. x ph <-> A. y ps )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   A.wal 1481   T. wtru 1484   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-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)