Theorem ralnralall 4449

Description: A contradiction concerning restricted generalization for a nonempty set implies anything. (Contributed by Alexander van der Vekens, 4-Sep-2018.)

Assertion

Ref	Expression
ralnralall	⊢ (𝐴 ≠ ∅ → ((∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 ¬ 𝜑) → 𝜓))

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem ralnralall

Step	Hyp	Ref	Expression
1		r19.26 3095	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∧ ¬ 𝜑) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 ¬ 𝜑))
2		pm3.24 403	. . . . 5 ⊢ ¬ (𝜑 ∧ ¬ 𝜑)
3	2	bifal 1555	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝜑) ↔ ⊥)
4	3	ralbii 3092	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∧ ¬ 𝜑) ↔ ∀𝑥 ∈ 𝐴 ⊥)
5		r19.3rzv 4429	. . . 4 ⊢ (𝐴 ≠ ∅ → (⊥ ↔ ∀𝑥 ∈ 𝐴 ⊥))
6		falim 1556	. . . 4 ⊢ (⊥ → 𝜓)
7	5, 6	syl6bir 253	. . 3 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 ⊥ → 𝜓))
8	4, 7	syl5bi 241	. 2 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 (𝜑 ∧ ¬ 𝜑) → 𝜓))
9	1, 8	syl5bir 242	1 ⊢ (𝐴 ≠ ∅ → ((∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 ¬ 𝜑) → 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396  ⊥wfal 1551   ≠ wne 2943  ∀wral 3064  ∅c0 4256

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-9 2116  ax-12 2171  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-ne 2944  df-ral 3069  df-dif 3890  df-nul 4257

This theorem is referenced by: (None)