Theorem rzal 4509

Description: Vacuous quantification is always true. (Contributed by NM, 11-Mar-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) Avoid df-clel 2816, ax-8 2110. (Revised by GG, 2-Sep-2024.)

Assertion

Ref	Expression
rzal	⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 𝜑)

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rzal

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfcleq 2730	. . . 4 ⊢ (𝐴 = {𝑦 ∣ ⊥} ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑦 ∣ ⊥}))
2	1	biimpi 216	. . 3 ⊢ (𝐴 = {𝑦 ∣ ⊥} → ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑦 ∣ ⊥}))
3		df-clab 2715	. . . . . 6 ⊢ (𝑥 ∈ {𝑦 ∣ ⊥} ↔ [𝑥 / 𝑦]⊥)
4		sbv 2088	. . . . . 6 ⊢ ([𝑥 / 𝑦]⊥ ↔ ⊥)
5	3, 4	bitri 275	. . . . 5 ⊢ (𝑥 ∈ {𝑦 ∣ ⊥} ↔ ⊥)
6	5	bibi2i 337	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑦 ∣ ⊥}) ↔ (𝑥 ∈ 𝐴 ↔ ⊥))
7		nbfal 1555	. . . . 5 ⊢ (¬ 𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ↔ ⊥))
8		pm2.21 123	. . . . 5 ⊢ (¬ 𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 → 𝜑))
9	7, 8	sylbir 235	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ↔ ⊥) → (𝑥 ∈ 𝐴 → 𝜑))
10	6, 9	sylbi 217	. . 3 ⊢ ((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑦 ∣ ⊥}) → (𝑥 ∈ 𝐴 → 𝜑))
11	2, 10	sylg 1823	. 2 ⊢ (𝐴 = {𝑦 ∣ ⊥} → ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
12		dfnul4 4335	. . 3 ⊢ ∅ = {𝑦 ∣ ⊥}
13	12	eqeq2i 2750	. 2 ⊢ (𝐴 = ∅ ↔ 𝐴 = {𝑦 ∣ ⊥})
14		df-ral 3062	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
15	11, 13, 14	3imtr4i 292	1 ⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206  ∀wal 1538   = wceq 1540  ⊥wfal 1552  [wsb 2064   ∈ wcel 2108  {cab 2714  ∀wral 3061  ∅c0 4333

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-ral 3062  df-dif 3954  df-nul 4334

This theorem is referenced by:  rexn0  4511  ral0  4513  ralf0  4514  raaan  4517  raaanv  4518  raaan2  4521  iinrab2  5070  riinrab  5084  reusv2lem2  5399  cnvpo  6307  dffi3  9471  brdom3  10568  dedekind  11424  fimaxre2  12213  fiminre2  12216  mgm0  18669  sgrp0  18740  efgs1  19753  opnnei  23128  bddiblnc  25877  axcontlem12  28990  nbgr0edg  29374  prcliscplgr  29431  cplgr0v  29444  0vtxrgr  29594  0vconngr  30212  frgr1v  30290  ubthlem1  30889  rdgssun  37379  matunitlindf  37625  mbfresfi  37673  blbnd  37794  rrnequiv  37842  upbdrech2  45320  limsupubuz  45728  stoweidlem9  46024  fourierdlem31  46153  upwordnul  46895  upwordsing  46899  0funcg2  48917