Theorem 19.43 767

Description: Theorem 19.43 of [Margaris] p. 90.

Assertion

Ref	Expression
19.43	|- (E.x(ph \/ ps) <-> (E.xph \/ E.xps))

Proof of Theorem 19.43

Step	Hyp	Ref	Expression
1		ioran 254	. . . . 5 |- (-. (ph \/ ps) <-> (-. ph /\ -. ps))
2	1	bial 695	. . . 4 |- (A.x -. (ph \/ ps) <-> A.x(-. ph /\ -. ps))
3		19.26 749	. . . 4 |- (A.x(-. ph /\ -. ps) <-> (A.x -. ph /\ A.x -. ps))
4		alnex 716	. . . . 5 |- (A.x -. ph <-> -. E.xph)
5		alnex 716	. . . . 5 |- (A.x -. ps <-> -. E.xps)
6	4, 5	anbi12i 369	. . . 4 |- ((A.x -. ph /\ A.x -. ps) <-> (-. E.xph /\ -. E.xps))
7	2, 3, 6	3bitr 155	. . 3 |- (A.x -. (ph \/ ps) <-> (-. E.xph /\ -. E.xps))
8	7	negbii 162	. 2 |- (-. A.x -. (ph \/ ps) <-> -. (-. E.xph /\ -. E.xps))
9		df-ex 679	. 2 |- (E.x(ph \/ ps) <-> -. A.x -. (ph \/ ps))
10		oran 255	. 2 |- ((E.xph \/ E.xps) <-> -. (-. E.xph /\ -. E.xps))
11	8, 9, 10	3bitr4 158	1 |- (E.x(ph \/ ps) <-> (E.xph \/ E.xps))

Syntax hints:  -. wn 1   <-> wb 127   \/ wo 195   /\ wa 196  A.wal 672  E.wex 678

This theorem is referenced by:  19.44 768  19.45 769  19.34 772  r19.43 1304  zfpair 1891  unpr 1930  uniun 1934  iunxun 2035  unopab 2121  dmun 2536  kmlem16 3595

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-or 197  df-an 198  df-ex 679

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