Theorem 2ralsng 4020

Description: Substitution expressed in terms of two quantifications over singletons. (Contributed by AV, 22-Dec-2019.)

Hypotheses

Ref	Expression
ralsng.1	|- ( x = A -> ( ph <-> ps ) )
2ralsng.1	|- ( y = B -> ( ps <-> ch ) )

Assertion

Ref	Expression
2ralsng	|- ( ( A e. V /\ B e. W ) -> ( A. x e. { A } A. y e. { B } ph <-> ch ) )

Distinct variable groups:   x, A   ps, x   y, A   x, B, y   ch, y

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

Proof of Theorem 2ralsng

Step	Hyp	Ref	Expression
1		ralsng.1	. . . 4 |- ( x = A -> ( ph <-> ps ) )
2	1	ralbidv 2839	. . 3 |- ( x = A -> ( A. y e. { B } ph <-> A. y e. { B } ps ) )
3	2	ralsng 4018	. 2 |- ( A e. V -> ( A. x e. { A } A. y e. { B } ph <-> A. y e. { B } ps ) )
4		2ralsng.1	. . 3 |- ( y = B -> ( ps <-> ch ) )
5	4	ralsng 4018	. 2 |- ( B e. W -> ( A. y e. { B } ps <-> ch ) )
6	3, 5	sylan9bb 711	1 |- ( ( A e. V /\ B e. W ) -> ( A. x e. { A } A. y e. { B } ph <-> ch ) )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 375   = wceq 1455   e. wcel 1898   A.wral 2749   {csn 3980

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-12 1944  ax-13 2102  ax-ext 2442

This theorem depends on definitions:  df-bi 190  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458  df-ral 2754  df-v 3059  df-sbc 3280  df-sn 3981

This theorem is referenced by:  mat1ghm  19557  mat1mhm  19558  c0snmgmhm  40187  zrrnghm  40190