Theorem albi 1107

Description: Split a biconditional and distribute quantifier.

Assertion

Ref	Expression
albi	|- (A.x(ph <-> ps) <-> (A.x(ph -> ps) /\ A.x(ps -> ph)))

Proof of Theorem albi

Step	Hyp	Ref	Expression
1		dfbi2 514	. . 3 |- ((ph <-> ps) <-> ((ph -> ps) /\ (ps -> ph)))
2	1	albii 999	. 2 |- (A.x(ph <-> ps) <-> A.x((ph -> ps) /\ (ps -> ph)))
3		19.26 1067	. 2 |- (A.x((ph -> ps) /\ (ps -> ph)) <-> (A.x(ph -> ps) /\ A.x(ps -> ph)))
4	2, 3	bitr 173	1 |- (A.x(ph <-> ps) <-> (A.x(ph -> ps) /\ A.x(ps -> ph)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 954

This theorem is referenced by:  2albi 1108  hbbid 1112  eu1 1392  eqss 2077  ssext 2763  dmcosseq 3365  homcard 10539

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

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