Theorem 19.41 1095

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 981	. 2 |- (E.x(ph /\ ps) <-> -. A.x -. (ph /\ ps))
2		19.41.1	. . . . . 6 |- (ps -> A.xps)
3	2	hbn 1004	. . . . 5 |- (-. ps -> A.x -. ps)
4	3	19.31 1087	. . . 4 |- (A.x(-. ph \/ -. ps) <-> (A.x -. ph \/ -. ps))
5		ianor 305	. . . . 5 |- (-. (ph /\ ps) <-> (-. ph \/ -. ps))
6	5	albii 999	. . . 4 |- (A.x -. (ph /\ ps) <-> A.x(-. ph \/ -. ps))
7		ianor 305	. . . . 5 |- (-. (E.xph /\ ps) <-> (-. E.xph \/ -. ps))
8		alnex 1033	. . . . . 6 |- (A.x -. ph <-> -. E.xph)
9	8	orbi1i 256	. . . . 5 |- ((A.x -. ph \/ -. ps) <-> (-. E.xph \/ -. ps))
10	7, 9	bitr4 176	. . . 4 |- (-. (E.xph /\ ps) <-> (A.x -. ph \/ -. ps))
11	4, 6, 10	3bitr4 183	. . 3 |- (A.x -. (ph /\ ps) <-> -. (E.xph /\ ps))
12	11	con2bii 221	. 2 |- ((E.xph /\ ps) <-> -. A.x -. (ph /\ ps))
13	1, 12	bitr4 176	1 |- (E.x(ph /\ ps) <-> (E.xph /\ ps))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223  A.wal 954  E.wex 980

This theorem is referenced by:  19.42 1096  sbf 1186  hbs1f 1189  19.41v 1305  eeanv 1323  euan 1428  2euex 1441  2exeu 1446  dfopab2 4113  dfoprab3 4114

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  ax-6o 978

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981

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