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Theorem ralrab 2998
 Description: Universal quantification over a restricted class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.)
Hypothesis
Ref Expression
ralab.1 (y = x → (φψ))
Assertion
Ref Expression
ralrab (x {y A φ}χx A (ψχ))
Distinct variable groups:   x,y   y,A   ψ,y
Allowed substitution hints:   φ(x,y)   ψ(x)   χ(x,y)   A(x)

Proof of Theorem ralrab
StepHypRef Expression
1 ralab.1 . . . . 5 (y = x → (φψ))
21elrab 2994 . . . 4 (x {y A φ} ↔ (x A ψ))
32imbi1i 315 . . 3 ((x {y A φ} → χ) ↔ ((x A ψ) → χ))
4 impexp 433 . . 3 (((x A ψ) → χ) ↔ (x A → (ψχ)))
53, 4bitri 240 . 2 ((x {y A φ} → χ) ↔ (x A → (ψχ)))
65ralbii2 2642 1 (x {y A φ}χx A (ψχ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∈ wcel 1710  ∀wral 2614  {crab 2618 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  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-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-rab 2623  df-v 2861 This theorem is referenced by: (None)
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