Theorem rzal 3652

Description: Vacuous quantification is always true. (Contributed by NM, 11-Mar-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

Ref	Expression
rzal	⊢ (A = ∅ → ∀x ∈ A φ)

Distinct variable group:   x,A

Allowed substitution hint:   φ(x)

Proof of Theorem rzal

Step	Hyp	Ref	Expression
1		ne0i 3557	. . . 4 ⊢ (x ∈ A → A ≠ ∅)
2	1	necon2bi 2563	. . 3 ⊢ (A = ∅ → ¬ x ∈ A)
3	2	pm2.21d 98	. 2 ⊢ (A = ∅ → (x ∈ A → φ))
4	3	ralrimiv 2697	1 ⊢ (A = ∅ → ∀x ∈ A φ)

Syntax hints:   → wi 4   = wceq 1642   ∈ wcel 1710  ∀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:  ralidm  3654  rgenz  3656  ralf0  3657  raaan  3658  raaanv  3659  iinrab2  4030  riinrab  4042