Theorem rzal 1773

Description: Vacuous quantification makes any wff true.

Assertion

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

Distinct variable group(s):   x,A

Proof of Theorem rzal

Step	Hyp	Ref	Expression
1		eleq2 1150	. . 3 |- (A = (/) -> (x e. A <-> x e. (/)))
2		noel 1711	. . . 4 |- -. x e. (/)
3	2	pm2.21i 73	. . 3 |- (x e. (/) -> ph)
4	1, 3	syl6bi 187	. 2 |- (A = (/) -> (x e. A -> ph))
5	4	r19.21aiv 1259	1 |- (A = (/) -> A.x e. A ph)

Syntax hints:   -> wi 2   = wceq 1091   e. wcel 1092  A.wral 1201  (/)c0 1707

This theorem is referenced by:  ralidm 1774  raaan 1775  chocnul 5293

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-v 1349  df-dif 1489  df-nul 1708

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