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Theorem ralab 2997
 Description: Universal quantification over a class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.)
Hypothesis
Ref Expression
ralab.1 (y = x → (φψ))
Assertion
Ref Expression
ralab (x {y φ}χx(ψχ))
Distinct variable groups:   x,y   ψ,y
Allowed substitution hints:   φ(x,y)   ψ(x)   χ(x,y)

Proof of Theorem ralab
StepHypRef Expression
1 df-ral 2619 . 2 (x {y φ}χx(x {y φ} → χ))
2 vex 2862 . . . . 5 x V
3 ralab.1 . . . . 5 (y = x → (φψ))
42, 3elab 2985 . . . 4 (x {y φ} ↔ ψ)
54imbi1i 315 . . 3 ((x {y φ} → χ) ↔ (ψχ))
65albii 1566 . 2 (x(x {y φ} → χ) ↔ x(ψχ))
71, 6bitri 240 1 (x {y φ}χx(ψχ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540   ∈ wcel 1710  {cab 2339  ∀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-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-v 2861 This theorem is referenced by:  nnadjoinpw  4521  funcnvuni  5161  frds  5935
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