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Theorem abbi 2134
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 2017 . 2 ({xφ} = {xψ} ↔ y(y {xφ} ↔ y {xψ}))
2 nfsab1 2013 . . . 4 x y {xφ}
3 nfsab1 2013 . . . 4 x y {xψ}
42, 3nfbi 1465 . . 3 x(y {xφ} ↔ y {xψ})
5 nfv 1403 . . 3 y(φψ)
6 df-clab 2010 . . . . 5 (y {xφ} ↔ [y / x]φ)
7 sbequ12r 1638 . . . . 5 (y = x → ([y / x]φφ))
86, 7syl5bb 181 . . . 4 (y = x → (y {xφ} ↔ φ))
9 df-clab 2010 . . . . 5 (y {xψ} ↔ [y / x]ψ)
10 sbequ12r 1638 . . . . 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 1620 . 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 1375  [wsb 1628  {cab 2009
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 1364  ax-ie2 1365  ax-8 1377  ax-11 1379  ax-4 1382  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411  ax-ext 2005
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1629  df-clab 2010  df-cleq 2016
This theorem is referenced by:  abbii  2136  abbid  2137  rabbi  2464  dfiota2  4793  iotabi  4801  uniabio  4802
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