Theorem r19.2zr 14295

Description: Quantifying a hypothesis with a universal restricted quantifier.

Hypothesis

Ref	Expression
r19.2zr.1	|- (ph -> ps)

Assertion

Ref	Expression
r19.2zr	|- ((A =/= (/) /\ A.x e. A ph) -> ps)

Distinct variable groups:   x,A   ps,x

Proof of Theorem r19.2zr

Step	Hyp	Ref	Expression
1		r19.2z 2958	. 2 |- ((A =/= (/) /\ A.x e. A ph) -> E.x e. A ph)
2		r19.2zr.1	. . 3 |- (ph -> ps)
3	2	r19.23aivr 14294	. 2 |- (E.x e. A ph -> ps)
4	1, 3	syl 12	1 |- ((A =/= (/) /\ A.x e. A ph) -> ps)

Syntax hints:   -> wi 3   /\ wa 240   =/= wne 2017  A.wral 2105  E.wrex 2106  (/)c0 2875

This theorem is referenced by:  r19.2zrr 14296

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-nul 2876

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