Theorem rzal 3589

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 3498	. . . 4 ⊢ (𝑥 ∈ 𝐴 → 𝐴 ≠ ∅)
2	1	necon2bi 2455	. . 3 ⊢ (𝐴 = ∅ → ¬ 𝑥 ∈ 𝐴)
3	2	pm2.21d 622	. 2 ⊢ (𝐴 = ∅ → (𝑥 ∈ 𝐴 → 𝜑))
4	3	ralrimiv 2602	1 ⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 𝜑)

Syntax hints:   → wi 4   = wceq 1395   ∈ wcel 2200  ∀wral 2508  ∅c0 3491

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-v 2801  df-dif 3199  df-nul 3492

This theorem is referenced by:  ralf0  3594  fiubm  11058  mgm0  13410  sgrp0  13451