Theorem rzal 3512

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 3421	. . . 4 |- ( x e. A -> A =/= (/) )
2	1	necon2bi 2395	. . 3 |- ( A = (/) -> -. x e. A )
3	2	pm2.21d 614	. 2 |- ( A = (/) -> ( x e. A -> ph ) )
4	3	ralrimiv 2542	1 |- ( A = (/) -> A. x e. A ph )

Syntax hints:   -> wi 4   = wceq 1348   e. wcel 2141   A.wral 2448   (/)c0 3414

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-v 2732  df-dif 3123  df-nul 3415

This theorem is referenced by:  ralf0  3518  fiubm  10763  mgm0  12623  sgrp0  12650