Theorem rgenzOLD 4077

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 4073	. 2 ⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 𝜑)
2		rgenzOLD.1	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ 𝑥 ∈ 𝐴) → 𝜑)
3	2	ralrimiva 2966	. 2 ⊢ (𝐴 ≠ ∅ → ∀𝑥 ∈ 𝐴 𝜑)
4	1, 3	pm2.61ine 2877	1 ⊢ ∀𝑥 ∈ 𝐴 𝜑

Syntax hints:   → wi 4   ∧ wa 384   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  ∅c0 3915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-v 3202  df-dif 3577  df-nul 3916

This theorem is referenced by: (None)