Theorem wl-sbcom2d-lem2 33343

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

Assertion

Ref	Expression
wl-sbcom2d-lem2	|- ( -. A. y y = x -> ( [ u / x ] [ v / y ] ph <-> A. x A. y ( ( x = u /\ y = v ) -> ph ) ) )

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

Allowed substitution hints:   ph( x, y)

Proof of Theorem wl-sbcom2d-lem2

Step	Hyp	Ref	Expression
1		id 22	. 2 |- ( -. A. y y = x -> -. A. y y = x )
2		wl-naev 33302	. 2 |- ( -. A. y y = x -> -. A. y y = v )
3		wl-naev 33302	. 2 |- ( -. A. y y = x -> -. A. y y = u )
4		wl-naev 33302	. 2 |- ( -. A. y y = x -> -. A. x x = u )
5	1, 2, 3, 4	wl-2sb6d 33341	1 |- ( -. A. y y = x -> ( [ u / x ] [ v / y ] ph <-> A. x A. y ( ( x = u /\ y = v ) -> 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-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  df-sb 1881

This theorem is referenced by:  wl-sbcom2d  33344