Theorem rgenzOLD 4029

Description: Obsolete as of 22-Jul-2021. (Contributed by NM, 8-Dec-2012.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
rgenzOLD.1	⊢ ((𝐴 ≠ ∅ ∧ 𝑥 ∈ 𝐴) → 𝜑)

Assertion

Ref	Expression
rgenzOLD	⊢ ∀𝑥 ∈ 𝐴 𝜑

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rgenzOLD

Step	Hyp	Ref	Expression
1		rzal 4025	. 2 ⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 𝜑)
2		rgenzOLD.1	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ 𝑥 ∈ 𝐴) → 𝜑)
3	2	ralrimiva 2949	. 2 ⊢ (𝐴 ≠ ∅ → ∀𝑥 ∈ 𝐴 𝜑)
4	1, 3	pm2.61ine 2865	1 ⊢ ∀𝑥 ∈ 𝐴 𝜑

Syntax hints:   → wi 4   ∧ wa 383   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∅c0 3874

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-v 3175  df-dif 3543  df-nul 3875

This theorem is referenced by: (None)