Theorem bj-dvelimdv1 32835

Description: Curried (exported) form of bj-dvelimdv 32834. (Contributed by BJ, 20-Oct-2021.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
bj-dvelimdv.nf	|- F/ x ph
bj-dvelimdv.nf1	|- ( ph -> F/ x ch )
bj-dvelimdv.is	|- ( z = y -> ( ch <-> ps ) )

Assertion

Ref	Expression
bj-dvelimdv1	|- ( ph -> ( -. A. x x = y -> F/ x ps ) )

Distinct variable groups:   x, z   y, z   ph, z   ps, z

Allowed substitution hints:   ph( x, y)   ps( x, y)   ch( x, y, z)

Proof of Theorem bj-dvelimdv1

Step	Hyp	Ref	Expression
1		nfeqf2 2297	. . . 4 |- ( -. A. x x = y -> F/ x z = y )
2		bj-dvelimdv.nf1	. . . 4 |- ( ph -> F/ x ch )
3		nfimt 1821	. . . 4 |- ( F/ x z = y -> ( F/ x ch -> F/ x ( z = y -> ch ) ) )
4	1, 2, 3	syl2imc 41	. . 3 |- ( ph -> ( -. A. x x = y -> F/ x ( z = y -> ch ) ) )
5	4	alrimdv 1857	. 2 |- ( ph -> ( -. A. x x = y -> A. z F/ x ( z = y -> ch ) ) )
6		bj-nfalt 32702	. 2 |- ( A. z F/ x ( z = y -> ch ) -> F/ x A. z ( z = y -> ch ) )
7		bj-dvelimdv.is	. . . 4 |- ( z = y -> ( ch <-> ps ) )
8	7	equsalvw 1931	. . 3 |- ( A. z ( z = y -> ch ) <-> ps )
9	8	nfbii 1778	. 2 |- ( F/ x A. z ( z = y -> ch ) <-> F/ x ps )
10	5, 6, 9	bj-syl66ib 32833	1 |- ( ph -> ( -. A. x x = y -> F/ x ps ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   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-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-ex 1705  df-nf 1710

This theorem is referenced by:  bj-dvelimv  32836  bj-axc14nf  32838