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Theorem ralrab2 2700
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 Aφ}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 2309 . . 3 {y Aφ} = {y ∣ (y A φ)}
21raleqi 2503 . 2 (x {y Aφ}ψx {y ∣ (y A φ)}ψ)
3 ralab2.1 . . 3 (x = y → (ψχ))
43ralab2 2699 . 2 (x {y ∣ (y A φ)}ψy((y A φ) → χ))
5 impexp 250 . . . 4 (((y A φ) → χ) ↔ (y A → (φχ)))
65albii 1356 . . 3 (y((y A φ) → χ) ↔ y(y A → (φχ)))
7 df-ral 2305 . . 3 (y A (φχ) ↔ y(y A → (φχ)))
86, 7bitr4i 176 . 2 (y((y A φ) → χ) ↔ y A (φχ))
92, 4, 83bitri 195 1 (x {y Aφ}ψy A (φχ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1240   wcel 1390  {cab 2023  wral 2300  {crab 2304
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rab 2309
This theorem is referenced by: (None)
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