Theorem 19.41 1136

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 1022	. 2 |- (E.x(ph /\ ps) <-> -. A.x -. (ph /\ ps))
2		19.41.1	. . . . . 6 |- (ps -> A.xps)
3	2	hbn 1045	. . . . 5 |- (-. ps -> A.x -. ps)
4	3	19.31 1128	. . . 4 |- (A.x(-. ph \/ -. ps) <-> (A.x -. ph \/ -. ps))
5		ianor 312	. . . . 5 |- (-. (ph /\ ps) <-> (-. ph \/ -. ps))
6	5	albii 1040	. . . 4 |- (A.x -. (ph /\ ps) <-> A.x(-. ph \/ -. ps))
7		ianor 312	. . . . 5 |- (-. (E.xph /\ ps) <-> (-. E.xph \/ -. ps))
8		alnex 1074	. . . . . 6 |- (A.x -. ph <-> -. E.xph)
9	8	orbi1i 263	. . . . 5 |- ((A.x -. ph \/ -. ps) <-> (-. E.xph \/ -. ps))
10	7, 9	bitr4i 183	. . . 4 |- (-. (E.xph /\ ps) <-> (A.x -. ph \/ -. ps))
11	4, 6, 10	3bitr4i 190	. . 3 |- (A.x -. (ph /\ ps) <-> -. (E.xph /\ ps))
12	11	con2bii 228	. 2 |- ((E.xph /\ ps) <-> -. A.x -. (ph /\ ps))
13	1, 12	bitr4i 183	1 |- (E.x(ph /\ ps) <-> (E.xph /\ ps))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 153   \/ wo 229   /\ wa 230  A.wal 995  E.wex 1021

This theorem is referenced by:  19.42 1137  sbf 1228  hbs1f 1231  19.41v 1347  eeanv 1365  euan 1470  2euex 1484  2exeu 1489  dfopab2 4171  dfoprab3 4172

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 1004  ax-4 1014  ax-5o 1016  ax-6o 1019

This theorem depends on definitions:  df-bi 154  df-or 231  df-an 232  df-ex 1022

Copyright terms: Public domain