Theorem ralnralall 4113

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 3093	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∧ ¬ 𝜑) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 ¬ 𝜑))
2		pm3.24 944	. . . . 5 ⊢ ¬ (𝜑 ∧ ¬ 𝜑)
3	2	bifal 1537	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝜑) ↔ ⊥)
4	3	ralbii 3009	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∧ ¬ 𝜑) ↔ ∀𝑥 ∈ 𝐴 ⊥)
5		r19.3rzv 4097	. . . 4 ⊢ (𝐴 ≠ ∅ → (⊥ ↔ ∀𝑥 ∈ 𝐴 ⊥))
6		falim 1538	. . . 4 ⊢ (⊥ → 𝜓)
7	5, 6	syl6bir 244	. . 3 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 ⊥ → 𝜓))
8	4, 7	syl5bi 232	. 2 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 (𝜑 ∧ ¬ 𝜑) → 𝜓))
9	1, 8	syl5bir 233	1 ⊢ (𝐴 ≠ ∅ → ((∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 ¬ 𝜑) → 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383  ⊥wfal 1528   ≠ wne 2823  ∀wral 2941  ∅c0 3948

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-v 3233  df-dif 3610  df-nul 3949

This theorem is referenced by: (None)