Theorem 19.35 1116

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 1108	. . . 4 |- (A.x(ph /\ -. ps) <-> (A.xph /\ A.x -. ps))
2		annim 245	. . . . 5 |- ((ph /\ -. ps) <-> -. (ph -> ps))
3	2	albii 1040	. . . 4 |- (A.x(ph /\ -. ps) <-> A.x -. (ph -> ps))
4		df-an 232	. . . 4 |- ((A.xph /\ A.x -. ps) <-> -. (A.xph -> -. A.x -. ps))
5	1, 3, 4	3bitr3i 188	. . 3 |- (A.x -. (ph -> ps) <-> -. (A.xph -> -. A.x -. ps))
6	5	con2bii 228	. 2 |- ((A.xph -> -. A.x -. ps) <-> -. A.x -. (ph -> ps))
7		df-ex 1022	. . 3 |- (E.xps <-> -. A.x -. ps)
8	7	imbi2i 192	. 2 |- ((A.xph -> E.xps) <-> (A.xph -> -. A.x -. ps))
9		df-ex 1022	. 2 |- (E.x(ph -> ps) <-> -. A.x -. (ph -> ps))
10	6, 8, 9	3bitr4ri 191	1 |- (E.x(ph -> ps) <-> (A.xph -> E.xps))

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

This theorem is referenced by:  19.35i 1117  19.35ri 1118  19.36 1119  19.37 1121  19.39 1123  19.24 1124  19.25 1125  sbequi 1270  grothprim 8866

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

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

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