Theorem r19.2z 2958

Description: Theorem 19.2 of [Margaris] p. 89 with restricted quantifiers (compare 19.2 1377). The restricted version is valid only when the domain of quantification is not empty.

Assertion

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

Distinct variable group:   x,A

Proof of Theorem r19.2z

Step	Hyp	Ref	Expression
1		df-ral 2109	. . . 4 |- (A.x e. A ph <-> A.x(x e. A -> ph))
2		exintr 1475	. . . 4 |- (A.x(x e. A -> ph) -> (E.x x e. A -> E.x(x e. A /\ ph)))
3	1, 2	sylbi 216	. . 3 |- (A.x e. A ph -> (E.x x e. A -> E.x(x e. A /\ ph)))
4		n0 2884	. . 3 |- (A =/= (/) <-> E.x x e. A)
5		df-rex 2110	. . 3 |- (E.x e. A ph <-> E.x(x e. A /\ ph))
6	3, 4, 5	3imtr4g 612	. 2 |- (A.x e. A ph -> (A =/= (/) -> E.x e. A ph))
7	6	impcom 378	1 |- ((A =/= (/) /\ A.x e. A ph) -> E.x e. A ph)

Syntax hints:   -> wi 3   /\ wa 240  A.wal 1296   e. wcel 1300  E.wex 1326   =/= wne 2017  A.wral 2105  E.wrex 2106  (/)c0 2875

This theorem is referenced by:  r19.2zb 2959  intssuni 3240  trintss 3427  eufromeq2 3829  eufromeq6 3833  onopriun 5118  alephval2 6050  cfeq0 6062  cfsuc 6063  isgrp2i 9360  bnj905 12821  trintssOLD 13795  dfon2lem6 13854  r19.2zr 14295  inpreima2 14468  cexint2 14862  fgsb 14921  fgsb2 14925  lvsovso3 15040  opnnei 15417  uffixfr 15575  filbcmb 15776  incsequz 15815  isbnd3 15941  totbndbnd 15944  heiborlem41 15995  rrntotbnd 16022  unichnidl 16179

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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