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Theorem raaan 3657
 Description: Rearrange restricted quantifiers. (Contributed by NM, 26-Oct-2010.)
Hypotheses
Ref Expression
raaan.1 yφ
raaan.2 xψ
Assertion
Ref Expression
raaan (x A y A (φ ψ) ↔ (x A φ y A ψ))
Distinct variable group:   x,y,A
Allowed substitution hints:   φ(x,y)   ψ(x,y)

Proof of Theorem raaan
StepHypRef Expression
1 rzal 3651 . . 3 (A = x A y A (φ ψ))
2 rzal 3651 . . 3 (A = x A φ)
3 rzal 3651 . . 3 (A = y A ψ)
4 pm5.1 830 . . 3 ((x A y A (φ ψ) (x A φ y A ψ)) → (x A y A (φ ψ) ↔ (x A φ y A ψ)))
51, 2, 3, 4syl12anc 1180 . 2 (A = → (x A y A (φ ψ) ↔ (x A φ y A ψ)))
6 raaan.1 . . . . 5 yφ
76r19.28z 3642 . . . 4 (A → (y A (φ ψ) ↔ (φ y A ψ)))
87ralbidv 2634 . . 3 (A → (x A y A (φ ψ) ↔ x A (φ y A ψ)))
9 nfcv 2489 . . . . 5 xA
10 raaan.2 . . . . 5 xψ
119, 10nfral 2667 . . . 4 xy A ψ
1211r19.27z 3648 . . 3 (A → (x A (φ y A ψ) ↔ (x A φ y A ψ)))
138, 12bitrd 244 . 2 (A → (x A y A (φ ψ) ↔ (x A φ y A ψ)))
145, 13pm2.61ine 2592 1 (x A y A (φ ψ) ↔ (x A φ y A ψ))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358  Ⅎwnf 1544   = wceq 1642   ≠ wne 2516  ∀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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