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Theorem rgen2a 2680
 Description: Generalization rule for restricted quantification. Note that x and y needn't be distinct (and illustrates the use of dvelim 2016). (Contributed by NM, 23-Nov-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof modification is discouraged.
Hypothesis
Ref Expression
rgen2a.1 ((x A y A) → φ)
Assertion
Ref Expression
rgen2a x A y A φ
Distinct variable group:   y,A
Allowed substitution hints:   φ(x,y)   A(x)

Proof of Theorem rgen2a
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 eleq1 2413 . . . . . . . 8 (y = x → (y Ax A))
2 rgen2a.1 . . . . . . . . 9 ((x A y A) → φ)
32ex 423 . . . . . . . 8 (x A → (y Aφ))
41, 3syl6bi 219 . . . . . . 7 (y = x → (y A → (y Aφ)))
54pm2.43d 44 . . . . . 6 (y = x → (y Aφ))
65alimi 1559 . . . . 5 (y y = xy(y Aφ))
76a1d 22 . . . 4 (y y = x → (x Ay(y Aφ)))
8 eleq1 2413 . . . . . 6 (z = x → (z Ax A))
98dvelimv 1939 . . . . 5 y y = x → (x Ay x A))
103alimi 1559 . . . . 5 (y x Ay(y Aφ))
119, 10syl6 29 . . . 4 y y = x → (x Ay(y Aφ)))
127, 11pm2.61i 156 . . 3 (x Ay(y Aφ))
13 df-ral 2619 . . 3 (y A φy(y Aφ))
1412, 13sylibr 203 . 2 (x Ay A φ)
1514rgen 2679 1 x A y A φ
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358  ∀wal 1540   = wceq 1642   ∈ wcel 1710  ∀wral 2614 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-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-cleq 2346  df-clel 2349  df-ral 2619 This theorem is referenced by:  vfinnc  4471  ncfinraise  4481  isoid  5490  pw1fnf1o  5855  fce  6188
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