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Theorem abbi 2463
 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) ↔ {x φ} = {x ψ})

Proof of Theorem abbi
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 dfcleq 2347 . 2 ({x φ} = {x ψ} ↔ y(y {x φ} ↔ y {x ψ}))
2 nfsab1 2343 . . . 4 x y {x φ}
3 nfsab1 2343 . . . 4 x y {x ψ}
42, 3nfbi 1834 . . 3 x(y {x φ} ↔ y {x ψ})
5 nfv 1619 . . 3 y(φψ)
6 df-clab 2340 . . . . 5 (y {x φ} ↔ [y / x]φ)
7 sbequ12r 1920 . . . . 5 (y = x → ([y / x]φφ))
86, 7syl5bb 248 . . . 4 (y = x → (y {x φ} ↔ φ))
9 df-clab 2340 . . . . 5 (y {x ψ} ↔ [y / x]ψ)
10 sbequ12r 1920 . . . . 5 (y = x → ([y / x]ψψ))
119, 10syl5bb 248 . . . 4 (y = x → (y {x ψ} ↔ ψ))
128, 11bibi12d 312 . . 3 (y = x → ((y {x φ} ↔ y {x ψ}) ↔ (φψ)))
134, 5, 12cbval 1984 . 2 (y(y {x φ} ↔ y {x ψ}) ↔ x(φψ))
141, 13bitr2i 241 1 (x(φψ) ↔ {x φ} = {x ψ})
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176  ∀wal 1540   = wceq 1642  [wsb 1648   ∈ wcel 1710  {cab 2339 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346 This theorem is referenced by:  abbii  2465  abbid  2466  rabbi  2789  dfeu2  4333  dfiota2  4340  iotabi  4348  uniabio  4349  iotanul  4354
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