Theorem r19.3rzvb 10437

Description: A more general version of r19.3rzv 2348.

Hypothesis

Ref	Expression
r19.3rzvb.1	|- (ph -> A.xph)

Assertion

Ref	Expression
r19.3rzvb	|- (A =/= (/) -> (ph <-> A.x e. A ph))

Distinct variable group:   x,A

Proof of Theorem r19.3rzvb

Step	Hyp	Ref	Expression
1		ne0 2288	. . 3 |- (A =/= (/) <-> E.x x e. A)
2		biimt 731	. . 3 |- (E.x x e. A -> (ph <-> (E.x x e. A -> ph)))
3	1, 2	sylbi 199	. 2 |- (A =/= (/) -> (ph <-> (E.x x e. A -> ph)))
4		df-ral 1649	. . 3 |- (A.x e. A ph <-> A.x(x e. A -> ph))
5		r19.3rzvb.1	. . . 4 |- (ph -> A.xph)
6	5	19.23 1063	. . 3 |- (A.x(x e. A -> ph) <-> (E.x x e. A -> ph))
7	4, 6	bitr 173	. 2 |- (A.x e. A ph <-> (E.x x e. A -> ph))
8	3, 7	syl6bbr 538	1 |- (A =/= (/) -> (ph <-> A.x e. A ph))

Syntax hints:   -> wi 3   <-> wb 146  A.wal 954   e. wcel 958  E.wex 980   =/= wne 1585  A.wral 1645  (/)c0 2280

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-v 1812  df-dif 2049  df-nul 2281

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