Theorem r19.26m 1291

Description: Theorem 19.26 of [Margaris] p. 90 with mixed quantifiers.

Assertion

Ref	Expression
r19.26m	|- (A.x((x e. A -> ph) /\ (x e. B -> ps)) <-> (A.x e. A ph /\ A.x e. B ps))

Proof of Theorem r19.26m

Step	Hyp	Ref	Expression
1		19.26 749	. 2 |- (A.x((x e. A -> ph) /\ (x e. B -> ps)) <-> (A.x(x e. A -> ph) /\ A.x(x e. B -> ps)))
2		df-ral 1205	. . 3 |- (A.x e. A ph <-> A.x(x e. A -> ph))
3		df-ral 1205	. . 3 |- (A.x e. B ps <-> A.x(x e. B -> ps))
4	2, 3	anbi12i 369	. 2 |- ((A.x e. A ph /\ A.x e. B ps) <-> (A.x(x e. A -> ph) /\ A.x(x e. B -> ps)))
5	1, 4	bitr4 154	1 |- (A.x((x e. A -> ph) /\ (x e. B -> ps)) <-> (A.x e. A ph /\ A.x e. B ps))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  A.wal 672   e. wcel 1092  A.wral 1201

This theorem is referenced by:  tfrlem5 2953

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-gen 677

This theorem depends on definitions:  df-bi 128  df-an 198  df-ral 1205

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