Theorem 19.29 1071

Description: Theorem 19.29 of [Margaris] p. 90.

Assertion

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

Proof of Theorem 19.29

Step	Hyp	Ref	Expression
1		19.20 994	. . . . 5 |- (A.x(ph -> -. ps) -> (A.xph -> A.x -. ps))
2		alnex 1033	. . . . 5 |- (A.x -. ps <-> -. E.xps)
3	1, 2	syl6ib 212	. . . 4 |- (A.x(ph -> -. ps) -> (A.xph -> -. E.xps))
4	3	con3i 98	. . 3 |- (-. (A.xph -> -. E.xps) -> -. A.x(ph -> -. ps))
5		df-an 225	. . 3 |- ((A.xph /\ E.xps) <-> -. (A.xph -> -. E.xps))
6		exnal 1038	. . 3 |- (E.x -. (ph -> -. ps) <-> -. A.x(ph -> -. ps))
7	4, 5, 6	3imtr4 219	. 2 |- ((A.xph /\ E.xps) -> E.x -. (ph -> -. ps))
8		df-an 225	. . 3 |- ((ph /\ ps) <-> -. (ph -> -. ps))
9	8	exbii 1051	. 2 |- (E.x(ph /\ ps) <-> E.x -. (ph -> -. ps))
10	7, 9	sylibr 200	1 |- ((A.xph /\ E.xps) -> E.x(ph /\ ps))

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

This theorem is referenced by:  19.29r 1072  19.29x 1074  exan 1106  r19.29 1756

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

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

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