Theorem rzal 2355

Description: Vacuous quantification is always true.

Assertion

Ref	Expression
rzal	|- (A = (/) -> A.x e. A ph)

Distinct variable group:   x,A

Proof of Theorem rzal

Step	Hyp	Ref	Expression
1		eleq2 1535	. . 3 |- (A = (/) -> (x e. A <-> x e. (/)))
2		noel 2284	. . . 4 |- -. x e. (/)
3	2	pm2.21i 77	. . 3 |- (x e. (/) -> ph)
4	1, 3	syl6bi 214	. 2 |- (A = (/) -> (x e. A -> ph))
5	4	r19.21aiv 1713	1 |- (A = (/) -> A.x e. A ph)

Syntax hints:   -> wi 3   = wceq 956   e. wcel 958  A.wral 1645  (/)c0 2280

This theorem is referenced by:  ralidm 2357  ralf0 2359  raaan 2360  cnvpo 3522  brdom3 4801

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-ral 1649  df-v 1812  df-dif 2049  df-nul 2281

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