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Theorem rzal 3651
 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
StepHypRef Expression
1 ne0i 3556 . . . 4 (x AA)
21necon2bi 2562 . . 3 (A = → ¬ x A)
32pm2.21d 98 . 2 (A = → (x Aφ))
43ralrimiv 2696 1 (A = x A φ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1642   ∈ wcel 1710  ∀wral 2614  ∅c0 3550 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 2478  df-ne 2518  df-ral 2619  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-nul 3551 This theorem is referenced by:  ralidm  3653  rgenz  3655  ralf0  3656  raaan  3657  raaanv  3658  iinrab2  4029  riinrab  4041
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