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Theorem ralbida 2628
 Description: Formula-building rule for restricted universal quantifier (deduction rule). (Contributed by NM, 6-Oct-2003.)
Hypotheses
Ref Expression
ralbida.1 xφ
ralbida.2 ((φ x A) → (ψχ))
Assertion
Ref Expression
ralbida (φ → (x A ψx A χ))

Proof of Theorem ralbida
StepHypRef Expression
1 ralbida.1 . . 3 xφ
2 ralbida.2 . . . 4 ((φ x A) → (ψχ))
32pm5.74da 668 . . 3 (φ → ((x Aψ) ↔ (x Aχ)))
41, 3albid 1772 . 2 (φ → (x(x Aψ) ↔ x(x Aχ)))
5 df-ral 2619 . 2 (x A ψx(x Aψ))
6 df-ral 2619 . 2 (x A χx(x Aχ))
74, 5, 63bitr4g 279 1 (φ → (x A ψx A χ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  Ⅎwnf 1544   ∈ 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-11 1746 This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-ral 2619 This theorem is referenced by:  ralbidva  2630  ralbid  2632  2ralbida  2653  ralbi  2750
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