Theorem 19.35 754

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		19.26 749	. . . 4 |- (A.x(ph /\ -. ps) <-> (A.xph /\ A.x -. ps))
2		annim 206	. . . . 5 |- ((ph /\ -. ps) <-> -. (ph -> ps))
3	2	bial 695	. . . 4 |- (A.x(ph /\ -. ps) <-> A.x -. (ph -> ps))
4		df-an 198	. . . 4 |- ((A.xph /\ A.x -. ps) <-> -. (A.xph -> -. A.x -. ps))
5	1, 3, 4	3bitr3 156	. . 3 |- (A.x -. (ph -> ps) <-> -. (A.xph -> -. A.x -. ps))
6	5	bicon2i 194	. 2 |- ((A.xph -> -. A.x -. ps) <-> -. A.x -. (ph -> ps))
7		df-ex 679	. . 3 |- (E.xps <-> -. A.x -. ps)
8	7	imbi2i 160	. 2 |- ((A.xph -> E.xps) <-> (A.xph -> -. A.x -. ps))
9		df-ex 679	. 2 |- (E.x(ph -> ps) <-> -. A.x -. (ph -> ps))
10	6, 8, 9	3bitr4r 159	1 |- (E.x(ph -> ps) <-> (A.xph -> E.xps))

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127   /\ wa 196  A.wal 672  E.wex 678

This theorem is referenced by:  19.35i 755  19.35ri 756  19.36 757  19.37 759  19.39 761  19.24 762  19.25 763  sbequi 876

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

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