Theorem rzal 4025

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 3880	. . . 4 ⊢ (𝑥 ∈ 𝐴 → 𝐴 ≠ ∅)
2	1	necon2bi 2812	. . 3 ⊢ (𝐴 = ∅ → ¬ 𝑥 ∈ 𝐴)
3	2	pm2.21d 117	. 2 ⊢ (𝐴 = ∅ → (𝑥 ∈ 𝐴 → 𝜑))
4	3	ralrimiv 2948	1 ⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 𝜑)

Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∅c0 3874

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-v 3175  df-dif 3543  df-nul 3875

This theorem is referenced by:  ralidm  4027  rgenzOLD  4029  ralf0OLD  4031  raaan  4032  iinrab2  4519  riinrab  4532  reusv2lem2  4795  reusv2lem2OLD  4796  cnvpo  5590  dffi3  8220  brdom3  9231  dedekind  10079  fimaxre2  10848  efgs1  17971  opnnei  20734  axcontlem12  25655  ubthlem1  27110  nofulllem2  31102  matunitlindf  32577  mbfresfi  32626  bddiblnc  32650  blbnd  32756  rrnequiv  32804  upbdrech2  38463  fiminre2  38535  stoweidlem9  38902  fourierdlem31  39031  raaan2  39824  nbgr0edg  40579  0vtxrgr  40776