Theorem ralf0 3344

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 603	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 → 𝜑) → (¬ 𝜑 → ¬ 𝑥 ∈ 𝐴))
3	1, 2	mpi 15	. . . 4 ⊢ ((𝑥 ∈ 𝐴 → 𝜑) → ¬ 𝑥 ∈ 𝐴)
4	3	alimi 1384	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) → ∀𝑥 ¬ 𝑥 ∈ 𝐴)
5		df-ral 2353	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
6		eq0 3266	. . 3 ⊢ (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴)
7	4, 5, 6	3imtr4i 199	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → 𝐴 = ∅)
8		rzal 3338	. 2 ⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 𝜑)
9	7, 8	impbii 124	1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ 𝐴 = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 103  ∀wal 1282   = wceq 1284   ∈ wcel 1433  ∀wral 2348  ∅c0 3251

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-v 2603  df-dif 2975  df-nul 3252

This theorem is referenced by: (None)