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Theorem ralbidv2 2636
 Description: Formula-building rule for restricted universal quantifier (deduction rule). (Contributed by NM, 6-Apr-1997.)
Hypothesis
Ref Expression
ralbidv2.1 (φ → ((x Aψ) ↔ (x Bχ)))
Assertion
Ref Expression
ralbidv2 (φ → (x A ψx B χ))
Distinct variable group:   φ,x
Allowed substitution hints:   ψ(x)   χ(x)   A(x)   B(x)

Proof of Theorem ralbidv2
StepHypRef Expression
1 ralbidv2.1 . . 3 (φ → ((x Aψ) ↔ (x Bχ)))
21albidv 1625 . 2 (φ → (x(x Aψ) ↔ x(x Bχ)))
3 df-ral 2619 . 2 (x A ψx(x Aψ))
4 df-ral 2619 . 2 (x B χx(x Bχ))
52, 3, 43bitr4g 279 1 (φ → (x A ψx B χ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540   ∈ wcel 1710  ∀wral 2614 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616 This theorem depends on definitions:  df-bi 177  df-ral 2619 This theorem is referenced by:  ralss  3332
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