Theorem 2ralsng 4220

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 2986	. . 3 |- ( x = A -> ( A. y e. { B } ph <-> A. y e. { B } ps ) )
3	2	ralsng 4218	. 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 4218	. 2 |- ( B e. W -> ( A. y e. { B } ps <-> ch ) )
6	3, 5	sylan9bb 736	1 |- ( ( A e. V /\ B e. W ) -> ( A. x e. { A } A. y e. { B } ph <-> ch ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {csn 4177

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-v 3202  df-sbc 3436  df-sn 4178

This theorem is referenced by:  mat1ghm  20289  mat1mhm  20290  c0snmgmhm  41914  zrrnghm  41917