Theorem rzal 4453

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 = ∅ → ∀𝑥 ∈ 𝐴 𝜑)

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rzal

Step	Hyp	Ref	Expression
1		ne0i 4300	. . . 4 ⊢ (𝑥 ∈ 𝐴 → 𝐴 ≠ ∅)
2	1	necon2bi 3046	. . 3 ⊢ (𝐴 = ∅ → ¬ 𝑥 ∈ 𝐴)
3	2	pm2.21d 121	. 2 ⊢ (𝐴 = ∅ → (𝑥 ∈ 𝐴 → 𝜑))
4	3	ralrimiv 3181	1 ⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 𝜑)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2114  ∀wral 3138  ∅c0 4291

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-ne 3017  df-ral 3143  df-dif 3939  df-nul 4292

This theorem is referenced by:  ralidm  4455  raaan  4460  raaanv  4461  raaan2  4464  iinrab2  4992  riinrab  5006  reusv2lem2  5300  cnvpo  6138  dffi3  8895  brdom3  9950  dedekind  10803  fimaxre2  11586  mgm0  17866  sgrp0  17908  efgs1  18861  opnnei  21728  axcontlem12  26761  nbgr0edg  27139  prcliscplgr  27196  cplgr0v  27209  0vtxrgr  27358  0vconngr  27972  frgr1v  28050  ubthlem1  28647  rdgssun  34662  matunitlindf  34905  mbfresfi  34953  bddiblnc  34977  blbnd  35080  rrnequiv  35128  upbdrech2  41595  fiminre2  41666  limsupubuz  42014  stoweidlem9  42314  fourierdlem31  42443