Theorem ralf0 4457

Description: The quantification of a falsehood is vacuous when true. (Contributed by NM, 26-Nov-2005.) (Proof shortened by JJ, 14-Jul-2021.)

Hypothesis

Ref	Expression
ralf0.1	⊢ ¬ 𝜑

Assertion

Ref	Expression
ralf0	⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ 𝐴 = ∅)

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem ralf0

Step	Hyp	Ref	Expression
1		ralf0.1	. . . 4 ⊢ ¬ 𝜑
2		mtt 367	. . . 4 ⊢ (¬ 𝜑 → (¬ 𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 → 𝜑)))
3	1, 2	ax-mp 5	. . 3 ⊢ (¬ 𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 → 𝜑))
4	3	albii 1820	. 2 ⊢ (∀𝑥 ¬ 𝑥 ∈ 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
5		eq0 4308	. 2 ⊢ (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴)
6		df-ral 3143	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
7	4, 5, 6	3bitr4ri 306	1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ 𝐴 = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208  ∀wal 1535   = 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-11 2161  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-dif 3939  df-nul 4292

This theorem is referenced by: (None)