ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  rexab2 Structured version   GIF version

Theorem rexab2 2701
Description: Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.)
Hypothesis
Ref Expression
ralab2.1 (x = y → (ψχ))
Assertion
Ref Expression
rexab2 (x {yφ}ψy(φ χ))
Distinct variable groups:   x,y   χ,x   φ,x   ψ,y
Allowed substitution hints:   φ(y)   ψ(x)   χ(y)

Proof of Theorem rexab2
StepHypRef Expression
1 df-rex 2306 . 2 (x {yφ}ψx(x {yφ} ψ))
2 nfsab1 2027 . . . 4 y x {yφ}
3 nfv 1418 . . . 4 yψ
42, 3nfan 1454 . . 3 y(x {yφ} ψ)
5 nfv 1418 . . 3 x(φ χ)
6 eleq1 2097 . . . . 5 (x = y → (x {yφ} ↔ y {yφ}))
7 abid 2025 . . . . 5 (y {yφ} ↔ φ)
86, 7syl6bb 185 . . . 4 (x = y → (x {yφ} ↔ φ))
9 ralab2.1 . . . 4 (x = y → (ψχ))
108, 9anbi12d 442 . . 3 (x = y → ((x {yφ} ψ) ↔ (φ χ)))
114, 5, 10cbvex 1636 . 2 (x(x {yφ} ψ) ↔ y(φ χ))
121, 11bitri 173 1 (x {yφ}ψy(φ χ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wex 1378   wcel 1390  {cab 2023  wrex 2301
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-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-rex 2306
This theorem is referenced by:  rexrab2  2702
  Copyright terms: Public domain W3C validator