Theorem rzal 2359

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 1538	. . 3 |- (A = (/) -> (x e. A <-> x e. (/)))
2		noel 2287	. . . 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 1716	1 |- (A = (/) -> A.x e. A ph)

Syntax hints:   -> wi 3   = wceq 958   e. wcel 960  A.wral 1648  (/)c0 2283

This theorem is referenced by:  ralidm 2361  ralf0 2363  raaan 2364  cnvpo 3528  brdom3 4811

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ral 1652  df-v 1815  df-dif 2052  df-nul 2284

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