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Theorem ralf0 3656
 Description: The quantification of a falsehood is vacuous when true. (Contributed by NM, 26-Nov-2005.)
Hypothesis
Ref Expression
ralf0.1 ¬ φ
Assertion
Ref Expression
ralf0 (x A φA = )
Distinct variable group:   x,A
Allowed substitution hint:   φ(x)

Proof of Theorem ralf0
StepHypRef Expression
1 ralf0.1 . . . . 5 ¬ φ
2 con3 126 . . . . 5 ((x Aφ) → (¬ φ → ¬ x A))
31, 2mpi 16 . . . 4 ((x Aφ) → ¬ x A)
43alimi 1559 . . 3 (x(x Aφ) → x ¬ x A)
5 df-ral 2619 . . 3 (x A φx(x Aφ))
6 eq0 3564 . . 3 (A = x ¬ x A)
74, 5, 63imtr4i 257 . 2 (x A φA = )
8 rzal 3651 . 2 (A = x A φ)
97, 8impbii 180 1 (x A φA = )
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176  ∀wal 1540   = wceq 1642   ∈ wcel 1710  ∀wral 2614  ∅c0 3550 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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: (None)
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