Theorem wl-sbcom2d-lem1 33342

Description: Lemma used to prove wl-sbcom2d 33344. (Contributed by Wolf Lammen, 10-Aug-2019.) (New usage is discouraged.)

Assertion

Ref	Expression
wl-sbcom2d-lem1	|- ( ( u = y /\ v = w ) -> ( -. A. x x = w -> ( [ u / x ] [ v / z ] ph <-> [ y / x ] [ w / z ] ph ) ) )

Distinct variable groups:   v, u, x   y, u, v   w, u, v   z, u, v   ph, u, v

Allowed substitution hints:   ph( x, y, z, w)

Proof of Theorem wl-sbcom2d-lem1

Step	Hyp	Ref	Expression
1		nfna1 2029	. . . . . 6 |- F/ x -. A. x x = w
2		nfeqf2 2297	. . . . . 6 |- ( -. A. x x = w -> F/ x v = w )
3	1, 2	nfan1 2068	. . . . 5 |- F/ x ( -. A. x x = w /\ v = w )
4		sbequ 2376	. . . . . 6 |- ( v = w -> ( [ v / z ] ph <-> [ w / z ] ph ) )
5	4	adantl 482	. . . . 5 |- ( ( -. A. x x = w /\ v = w ) -> ( [ v / z ] ph <-> [ w / z ] ph ) )
6	3, 5	sbbid 2403	. . . 4 |- ( ( -. A. x x = w /\ v = w ) -> ( [ u / x ] [ v / z ] ph <-> [ u / x ] [ w / z ] ph ) )
7	6	ancoms 469	. . 3 |- ( ( v = w /\ -. A. x x = w ) -> ( [ u / x ] [ v / z ] ph <-> [ u / x ] [ w / z ] ph ) )
8		sbequ 2376	. . 3 |- ( u = y -> ( [ u / x ] [ w / z ] ph <-> [ y / x ] [ w / z ] ph ) )
9	7, 8	sylan9bbr 737	. 2 |- ( ( u = y /\ ( v = w /\ -. A. x x = w ) ) -> ( [ u / x ] [ v / z ] ph <-> [ y / x ] [ w / z ] ph ) )
10	9	expr 643	1 |- ( ( u = y /\ v = w ) -> ( -. A. x x = w -> ( [ u / x ] [ v / z ] ph <-> [ y / x ] [ w / z ] ph ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   [wsb 1880

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-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  df-sb 1881

This theorem is referenced by:  wl-sbcom2d  33344