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Theorem ralrab2 3002
 Description: Universal quantification over a restricted class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.)
Hypothesis
Ref Expression
ralab2.1 (x = y → (ψχ))
Assertion
Ref Expression
ralrab2 (x {y A φ}ψy A (φχ))
Distinct variable groups:   x,y   x,A   χ,x   φ,x   ψ,y
Allowed substitution hints:   φ(y)   ψ(x)   χ(y)   A(y)

Proof of Theorem ralrab2
StepHypRef Expression
1 df-rab 2623 . . 3 {y A φ} = {y (y A φ)}
21raleqi 2811 . 2 (x {y A φ}ψx {y (y A φ)}ψ)
3 ralab2.1 . . 3 (x = y → (ψχ))
43ralab2 3001 . 2 (x {y (y A φ)}ψy((y A φ) → χ))
5 impexp 433 . . . 4 (((y A φ) → χ) ↔ (y A → (φχ)))
65albii 1566 . . 3 (y((y A φ) → χ) ↔ y(y A → (φχ)))
7 df-ral 2619 . . 3 (y A (φχ) ↔ y(y A → (φχ)))
86, 7bitr4i 243 . 2 (y((y A φ) → χ) ↔ y A (φχ))
92, 4, 83bitri 262 1 (x {y A φ}ψy A (φχ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540   ∈ wcel 1710  {cab 2339  ∀wral 2614  {crab 2618 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-rab 2623 This theorem is referenced by: (None)
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