Theorem r2alf 2938

Description: Double restricted universal quantification. (Contributed by Mario Carneiro, 14-Oct-2016.) Use r2allem 2937. (Revised by Wolf Lammen, 9-Jan-2020.)

Hypothesis

Ref	Expression
r2alf.1	|- F/_ y A

Assertion

Ref	Expression
r2alf	|- ( A. x e. A A. y e. B ph <-> A. x A. y ( ( x e. A /\ y e. B ) -> ph ) )

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)   A( x, y)   B( x, y)

Proof of Theorem r2alf

Step	Hyp	Ref	Expression
1		r2alf.1	. . . 4 |- F/_ y A
2	1	nfcri 2758	. . 3 |- F/ y x e. A
3	2	19.21 2075	. 2 |- ( A. y ( x e. A -> ( y e. B -> ph ) ) <-> ( x e. A -> A. y ( y e. B -> ph ) ) )
4	3	r2allem 2937	1 |- ( A. x e. A A. y e. B ph <-> A. x A. y ( ( x e. A /\ y e. B ) -> ph ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   e. wcel 1990   F/_wnfc 2751   A.wral 2912

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-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917

This theorem is referenced by:  r2exf  3060  ralcomf  3096