Theorem rzal 3548

Description: Vacuous quantification is always true. (Contributed by NM, 11-Mar-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

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

Distinct variable group:   x, A

Allowed substitution hint:   ph( x)

Proof of Theorem rzal

Step	Hyp	Ref	Expression
1		ne0i 3457	. . . 4 |- ( x e. A -> A =/= (/) )
2	1	necon2bi 2422	. . 3 |- ( A = (/) -> -. x e. A )
3	2	pm2.21d 620	. 2 |- ( A = (/) -> ( x e. A -> ph ) )
4	3	ralrimiv 2569	1 |- ( A = (/) -> A. x e. A ph )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2167   A.wral 2475   (/)c0 3450

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-v 2765  df-dif 3159  df-nul 3451

This theorem is referenced by:  ralf0  3553  fiubm  10920  mgm0  13012  sgrp0  13053