Theorem r19.26m 1752

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 1067	. 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 1649	. . 3 |- (A.x e. A ph <-> A.x(x e. A -> ph))
3		df-ral 1649	. . 3 |- (A.x e. B ps <-> A.x(x e. B -> ps))
4	2, 3	anbi12i 482	. 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 176	1 |- (A.x((x e. A -> ph) /\ (x e. B -> ps)) <-> (A.x e. A ph /\ A.x e. B ps))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 954   e. wcel 958  A.wral 1645

This theorem is referenced by:  tfrlem5 3915

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 963  ax-4 973  ax-5o 975

This theorem depends on definitions:  df-bi 147  df-an 225  df-ral 1649

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