Theorem ralf0 3564

Description: The quantification of a falsehood is vacuous when true. (Contributed by NM, 26-Nov-2005.)

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	. . . . 5 ⊢ ¬ 𝜑
2		con3 643	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 → 𝜑) → (¬ 𝜑 → ¬ 𝑥 ∈ 𝐴))
3	1, 2	mpi 15	. . . 4 ⊢ ((𝑥 ∈ 𝐴 → 𝜑) → ¬ 𝑥 ∈ 𝐴)
4	3	alimi 1479	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) → ∀𝑥 ¬ 𝑥 ∈ 𝐴)
5		df-ral 2490	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
6		eq0 3480	. . 3 ⊢ (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴)
7	4, 5, 6	3imtr4i 201	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → 𝐴 = ∅)
8		rzal 3559	. 2 ⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 𝜑)
9	7, 8	impbii 126	1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ 𝐴 = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 105  ∀wal 1371   = wceq 1373   ∈ wcel 2177  ∀wral 2485  ∅c0 3461

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-v 2775  df-dif 3169  df-nul 3462

This theorem is referenced by: (None)