Theorem 19.41 774

Description: Theorem 19.41 of [Margaris] p. 90.

Hypothesis

Ref	Expression
19.41.1	|- (ps -> A.xps)

Assertion

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

Proof of Theorem 19.41

Step	Hyp	Ref	Expression
1		df-ex 679	. 2 |- (E.x(ph /\ ps) <-> -. A.x -. (ph /\ ps))
2		19.41.1	. . . . . 6 |- (ps -> A.xps)
3	2	hbne 699	. . . . 5 |- (-. ps -> A.x -. ps)
4	3	19.31 766	. . . 4 |- (A.x(-. ph \/ -. ps) <-> (A.x -. ph \/ -. ps))
5		ianor 253	. . . . 5 |- (-. (ph /\ ps) <-> (-. ph \/ -. ps))
6	5	bial 695	. . . 4 |- (A.x -. (ph /\ ps) <-> A.x(-. ph \/ -. ps))
7		ianor 253	. . . . 5 |- (-. (E.xph /\ ps) <-> (-. E.xph \/ -. ps))
8		alnex 716	. . . . . 6 |- (A.x -. ph <-> -. E.xph)
9	8	orbi1i 215	. . . . 5 |- ((A.x -. ph \/ -. ps) <-> (-. E.xph \/ -. ps))
10	7, 9	bitr4 154	. . . 4 |- (-. (E.xph /\ ps) <-> (A.x -. ph \/ -. ps))
11	4, 6, 10	3bitr4 158	. . 3 |- (A.x -. (ph /\ ps) <-> -. (E.xph /\ ps))
12	11	bicon2i 194	. 2 |- ((E.xph /\ ps) <-> -. A.x -. (ph /\ ps))
13	1, 12	bitr4 154	1 |- (E.x(ph /\ ps) <-> (E.xph /\ ps))

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127   \/ wo 195   /\ wa 196  A.wal 672  E.wex 678

This theorem is referenced by:  19.42 775  sbf 870  hbs1f 874  19.41v 963  eeanv 980  2euex 1061  2exeu 1066

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  df-ex 679

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