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Theorem abbi 2133
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 2016 . 2 ({xφ} = {xψ} ↔ y(y {xφ} ↔ y {xψ}))
2 nfsab1 2012 . . . 4 x y {xφ}
3 nfsab1 2012 . . . 4 x y {xψ}
42, 3nfbi 1463 . . 3 x(y {xφ} ↔ y {xψ})
5 nfv 1402 . . 3 y(φψ)
6 df-clab 2009 . . . . 5 (y {xφ} ↔ [y / x]φ)
7 sbequ12r 1637 . . . . 5 (y = x → ([y / x]φφ))
86, 7syl5bb 181 . . . 4 (y = x → (y {xφ} ↔ φ))
9 df-clab 2009 . . . . 5 (y {xψ} ↔ [y / x]ψ)
10 sbequ12r 1637 . . . . 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 1619 . 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 1226   = wceq 1228   wcel 1374  [wsb 1627  {cab 2008
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 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-11 1378  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015
This theorem is referenced by:  abbii  2135  abbid  2136  rabbi  2463  dfiota2  4793  iotabi  4801  uniabio  4802
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