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Theorem raaan 3307
 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 raaan.1 . . . 4 yφ
2 raaan.2 . . . 4 xψ
31, 2raaanlem 3306 . . 3 (x x A → (x A y A (φ ψ) ↔ (x A φ y A ψ)))
43pm5.74i 169 . 2 ((x x Ax A y A (φ ψ)) ↔ (x x A → (x A φ y A ψ)))
5 ralm 3305 . 2 ((x x Ax A y A (φ ψ)) ↔ x A y A (φ ψ))
6 jcab 522 . . 3 ((x x A → (x A φ y A ψ)) ↔ ((x x Ax A φ) (x x Ay A ψ)))
7 ralm 3305 . . . 4 ((x x Ax A φ) ↔ x A φ)
8 eleq1 2082 . . . . . . 7 (x = y → (x Ay A))
98cbvexv 1777 . . . . . 6 (x x Ay y A)
109imbi1i 227 . . . . 5 ((x x Ay A ψ) ↔ (y y Ay A ψ))
11 ralm 3305 . . . . 5 ((y y Ay A ψ) ↔ y A ψ)
1210, 11bitri 173 . . . 4 ((x x Ay A ψ) ↔ y A ψ)
137, 12anbi12i 436 . . 3 (((x x Ax A φ) (x x Ay A ψ)) ↔ (x A φ y A ψ))
146, 13bitri 173 . 2 ((x x A → (x A φ y A ψ)) ↔ (x A φ y A ψ))
154, 5, 143bitr3i 199 1 (x A y A (φ ψ) ↔ (x A φ y A ψ))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  Ⅎwnf 1328  ∃wex 1361   ∈ wcel 1375  ∀wral 2282 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1315  ax-7 1316  ax-gen 1317  ax-ie1 1362  ax-ie2 1363  ax-8 1377  ax-4 1382  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411  ax-ext 2004 This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1329  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2287 This theorem is referenced by:  raaanv  3308
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