Theorem 19.35 1073

Description: Theorem 19.35 of [Margaris] p. 90. This theorem is useful for moving an implication (in the form of the right-hand side) into the scope of a single existential quantifier.

Assertion

Ref	Expression
19.35	|- (E.x(ph -> ps) <-> (A.xph -> E.xps))

Proof of Theorem 19.35

Step	Hyp	Ref	Expression
1		19.26 1065	. . . 4 |- (A.x(ph /\ -. ps) <-> (A.xph /\ A.x -. ps))
2		annim 238	. . . . 5 |- ((ph /\ -. ps) <-> -. (ph -> ps))
3	2	albii 997	. . . 4 |- (A.x(ph /\ -. ps) <-> A.x -. (ph -> ps))
4		df-an 225	. . . 4 |- ((A.xph /\ A.x -. ps) <-> -. (A.xph -> -. A.x -. ps))
5	1, 3, 4	3bitr3 181	. . 3 |- (A.x -. (ph -> ps) <-> -. (A.xph -> -. A.x -. ps))
6	5	con2bii 221	. 2 |- ((A.xph -> -. A.x -. ps) <-> -. A.x -. (ph -> ps))
7		df-ex 979	. . 3 |- (E.xps <-> -. A.x -. ps)
8	7	imbi2i 185	. 2 |- ((A.xph -> E.xps) <-> (A.xph -> -. A.x -. ps))
9		df-ex 979	. 2 |- (E.x(ph -> ps) <-> -. A.x -. (ph -> ps))
10	6, 8, 9	3bitr4r 184	1 |- (E.x(ph -> ps) <-> (A.xph -> E.xps))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223  A.wal 952  E.wex 978

This theorem is referenced by:  19.35i 1074  19.35ri 1075  19.36 1076  19.37 1078  19.39 1080  19.24 1081  19.25 1082  sbequi 1226  grothprim 8722

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 961  ax-4 971  ax-5o 973

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 979

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