Theorem ralnralall 4430

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 3092	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∧ ¬ 𝜑) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 ¬ 𝜑))
2		pm3.24 406	. . . . 5 ⊢ ¬ (𝜑 ∧ ¬ 𝜑)
3	2	bifal 1559	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝜑) ↔ ⊥)
4	3	ralbii 3088	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∧ ¬ 𝜑) ↔ ∀𝑥 ∈ 𝐴 ⊥)
5		r19.3rzv 4410	. . . 4 ⊢ (𝐴 ≠ ∅ → (⊥ ↔ ∀𝑥 ∈ 𝐴 ⊥))
6		falim 1560	. . . 4 ⊢ (⊥ → 𝜓)
7	5, 6	syl6bir 257	. . 3 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 ⊥ → 𝜓))
8	4, 7	syl5bi 245	. 2 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 (𝜑 ∧ ¬ 𝜑) → 𝜓))
9	1, 8	syl5bir 246	1 ⊢ (𝐴 ≠ ∅ → ((∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 ¬ 𝜑) → 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399  ⊥wfal 1555   ≠ wne 2940  ∀wral 3061  ∅c0 4237

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-9 2120  ax-12 2175  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-clab 2715  df-cleq 2729  df-ne 2941  df-ral 3066  df-dif 3869  df-nul 4238

This theorem is referenced by: (None)