Theorem ralnralall 4460

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 3172	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∧ ¬ 𝜑) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 ¬ 𝜑))
2		pm3.24 405	. . . . 5 ⊢ ¬ (𝜑 ∧ ¬ 𝜑)
3	2	bifal 1553	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝜑) ↔ ⊥)
4	3	ralbii 3167	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∧ ¬ 𝜑) ↔ ∀𝑥 ∈ 𝐴 ⊥)
5		r19.3rzv 4446	. . . 4 ⊢ (𝐴 ≠ ∅ → (⊥ ↔ ∀𝑥 ∈ 𝐴 ⊥))
6		falim 1554	. . . 4 ⊢ (⊥ → 𝜓)
7	5, 6	syl6bir 256	. . 3 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 ⊥ → 𝜓))
8	4, 7	syl5bi 244	. 2 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 (𝜑 ∧ ¬ 𝜑) → 𝜓))
9	1, 8	syl5bir 245	1 ⊢ (𝐴 ≠ ∅ → ((∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 ¬ 𝜑) → 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398  ⊥wfal 1549   ≠ wne 3018  ∀wral 3140  ∅c0 4293

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-fal 1550  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-dif 3941  df-nul 4294

This theorem is referenced by: (None)