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

Theorem abbi 2148
 Description: Equivalent wff's correspond to equal class abstractions. (Contributed by NM, 25-Nov-2013.) (Revised by Mario Carneiro, 11-Aug-2016.)
Assertion
Ref Expression
abbi (x(φψ) ↔ {xφ} = {xψ})

Proof of Theorem abbi
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 dfcleq 2031 . 2 ({xφ} = {xψ} ↔ y(y {xφ} ↔ y {xψ}))
2 nfsab1 2027 . . . 4 x y {xφ}
3 nfsab1 2027 . . . 4 x y {xψ}
42, 3nfbi 1478 . . 3 x(y {xφ} ↔ y {xψ})
5 nfv 1418 . . 3 y(φψ)
6 df-clab 2024 . . . . 5 (y {xφ} ↔ [y / x]φ)
7 sbequ12r 1652 . . . . 5 (y = x → ([y / x]φφ))
86, 7syl5bb 181 . . . 4 (y = x → (y {xφ} ↔ φ))
9 df-clab 2024 . . . . 5 (y {xψ} ↔ [y / x]ψ)
10 sbequ12r 1652 . . . . 5 (y = x → ([y / x]ψψ))
119, 10syl5bb 181 . . . 4 (y = x → (y {xψ} ↔ ψ))
128, 11bibi12d 224 . . 3 (y = x → ((y {xφ} ↔ y {xψ}) ↔ (φψ)))
134, 5, 12cbval 1634 . 2 (y(y {xφ} ↔ y {xψ}) ↔ x(φψ))
141, 13bitr2i 174 1 (x(φψ) ↔ {xφ} = {xψ})
 Colors of variables: wff set class Syntax hints:   ↔ wb 98  ∀wal 1240   = wceq 1242   ∈ wcel 1390  [wsb 1642  {cab 2023 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-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 This theorem is referenced by:  abbii  2150  abbid  2151  rabbi  2481  dfiota2  4811  iotabi  4819  uniabio  4820
 Copyright terms: Public domain W3C validator