Theorem rgenz 3656

Description: Generalization rule that eliminates a nonzero class requirement. (Contributed by NM, 8-Dec-2012.)

Hypothesis

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

Assertion

Ref	Expression
rgenz	⊢ ∀x ∈ A φ

Distinct variable group:   x,A

Allowed substitution hint:   φ(x)

Proof of Theorem rgenz

Step	Hyp	Ref	Expression
1		rzal 3652	. 2 ⊢ (A = ∅ → ∀x ∈ A φ)
2		rgenz.1	. . 3 ⊢ ((A ≠ ∅ ∧ x ∈ A) → φ)
3	2	ralrimiva 2698	. 2 ⊢ (A ≠ ∅ → ∀x ∈ A φ)
4	1, 3	pm2.61ine 2593	1 ⊢ ∀x ∈ A φ

Syntax hints:   → wi 4   ∧ wa 358   ∈ wcel 1710   ≠ wne 2517  ∀wral 2615  ∅c0 3551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-nul 3552

This theorem is referenced by: (None)