Theorem 19.32 765

Description: Theorem 19.32 of [Margaris] p. 90.

Hypothesis

Ref	Expression
19.32.1	|- (ph -> A.xph)

Assertion

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

Proof of Theorem 19.32

Step	Hyp	Ref	Expression
1		19.32.1	. . . 4 |- (ph -> A.xph)
2	1	hbne 699	. . 3 |- (-. ph -> A.x -. ph)
3	2	19.21 738	. 2 |- (A.x(-. ph -> ps) <-> (-. ph -> A.xps))
4		df-or 197	. . 3 |- ((ph \/ ps) <-> (-. ph -> ps))
5	4	bial 695	. 2 |- (A.x(ph \/ ps) <-> A.x(-. ph -> ps))
6		df-or 197	. 2 |- ((ph \/ A.xps) <-> (-. ph -> A.xps))
7	3, 5, 6	3bitr4 158	1 |- (A.x(ph \/ ps) <-> (ph \/ A.xps))

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127   \/ wo 195  A.wal 672

This theorem is referenced by:  19.31 766  2eu3 1069

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-6 675  ax-gen 677

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198

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