Theorem r19.2z 2347

Description: Theorem 19.2 of [Margaris] p. 89 with restricted quantifiers (compare 19.2 1030). 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 1649	. . . 4 |- (A.x e. A ph <-> A.x(x e. A -> ph))
2		exintr 1117	. . . 4 |- (A.x(x e. A -> ph) -> (E.x x e. A -> E.x(x e. A /\ ph)))
3	1, 2	sylbi 199	. . 3 |- (A.x e. A ph -> (E.x x e. A -> E.x(x e. A /\ ph)))
4		ne0 2288	. . 3 |- (A =/= (/) <-> E.x x e. A)
5		df-rex 1650	. . 3 |- (E.x e. A ph <-> E.x(x e. A /\ ph))
6	3, 4, 5	3imtr4g 553	. 2 |- (A.x e. A ph -> (A =/= (/) -> E.x e. A ph))
7	6	impcom 351	1 |- ((A =/= (/) /\ A.x e. A ph) -> E.x e. A ph)

Syntax hints:   -> wi 3   /\ wa 223  A.wal 954   e. wcel 958  E.wex 980   =/= wne 1585  A.wral 1645  E.wrex 1646  (/)c0 2280

This theorem is referenced by:  intssuni 2555  alephval2 4902  cfeq0 4914  cfsuc 4915  isgrp2i 8076  fgsb 10570  fgsbOLD 10571  fgsb2 10580

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-nul 2281

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