Theorem 19.29 752

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		19.20 690	. . . . 5 |- (A.x(ph -> -. ps) -> (A.xph -> A.x -. ps))
2		alnex 716	. . . . 5 |- (A.x -. ps <-> -. E.xps)
3	1, 2	syl6ib 185	. . . 4 |- (A.x(ph -> -. ps) -> (A.xph -> -. E.xps))
4	3	con3i 90	. . 3 |- (-. (A.xph -> -. E.xps) -> -. A.x(ph -> -. ps))
5		df-an 198	. . 3 |- ((A.xph /\ E.xps) <-> -. (A.xph -> -. E.xps))
6		exnal 721	. . 3 |- (E.x -. (ph -> -. ps) <-> -. A.x(ph -> -. ps))
7	4, 5, 6	3imtr4 192	. 2 |- ((A.xph /\ E.xps) -> E.x -. (ph -> -. ps))
8		df-an 198	. . 3 |- ((ph /\ ps) <-> -. (ph -> -. ps))
9	8	biex 733	. 2 |- (E.x(ph /\ ps) <-> E.x -. (ph -> -. ps))
10	7, 9	sylibr 175	1 |- ((A.xph /\ E.xps) -> E.x(ph /\ ps))

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

This theorem is referenced by:  19.29r 753  exan 784  r19.29 1295

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-an 198  df-ex 679

metamath.org