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Theorem 2ralbida 2339
 Description: Formula-building rule for restricted universal quantifier (deduction rule). (Contributed by NM, 24-Feb-2004.)
Hypotheses
Ref Expression
2ralbida.1 xφ
2ralbida.2 yφ
2ralbida.3 ((φ (x A y B)) → (ψχ))
Assertion
Ref Expression
2ralbida (φ → (x A y B ψx A y B χ))
Distinct variable groups:   x,y   y,A
Allowed substitution hints:   φ(x,y)   ψ(x,y)   χ(x,y)   A(x)   B(x,y)

Proof of Theorem 2ralbida
StepHypRef Expression
1 2ralbida.1 . 2 xφ
2 2ralbida.2 . . . 4 yφ
3 nfv 1418 . . . 4 y x A
42, 3nfan 1454 . . 3 y(φ x A)
5 2ralbida.3 . . . 4 ((φ (x A y B)) → (ψχ))
65anassrs 380 . . 3 (((φ x A) y B) → (ψχ))
74, 6ralbida 2314 . 2 ((φ x A) → (y B ψy B χ))
81, 7ralbida 2314 1 (φ → (x A y B ψx A y B χ))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  Ⅎwnf 1346   ∈ wcel 1390  ∀wral 2300 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 1333  ax-gen 1335  ax-4 1397  ax-17 1416 This theorem depends on definitions:  df-bi 110  df-nf 1347  df-ral 2305 This theorem is referenced by:  2ralbidva  2340
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