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Theorem abbi 2167
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 2050 . 2 ({𝑥𝜑} = {𝑥𝜓} ↔ ∀𝑦(𝑦 ∈ {𝑥𝜑} ↔ 𝑦 ∈ {𝑥𝜓}))
2 nfsab1 2046 . . . 4 𝑥 𝑦 ∈ {𝑥𝜑}
3 nfsab1 2046 . . . 4 𝑥 𝑦 ∈ {𝑥𝜓}
42, 3nfbi 1497 . . 3 𝑥(𝑦 ∈ {𝑥𝜑} ↔ 𝑦 ∈ {𝑥𝜓})
5 nfv 1437 . . 3 𝑦(𝜑𝜓)
6 df-clab 2043 . . . . 5 (𝑦 ∈ {𝑥𝜑} ↔ [𝑦 / 𝑥]𝜑)
7 sbequ12r 1671 . . . . 5 (𝑦 = 𝑥 → ([𝑦 / 𝑥]𝜑𝜑))
86, 7syl5bb 185 . . . 4 (𝑦 = 𝑥 → (𝑦 ∈ {𝑥𝜑} ↔ 𝜑))
9 df-clab 2043 . . . . 5 (𝑦 ∈ {𝑥𝜓} ↔ [𝑦 / 𝑥]𝜓)
10 sbequ12r 1671 . . . . 5 (𝑦 = 𝑥 → ([𝑦 / 𝑥]𝜓𝜓))
119, 10syl5bb 185 . . . 4 (𝑦 = 𝑥 → (𝑦 ∈ {𝑥𝜓} ↔ 𝜓))
128, 11bibi12d 228 . . 3 (𝑦 = 𝑥 → ((𝑦 ∈ {𝑥𝜑} ↔ 𝑦 ∈ {𝑥𝜓}) ↔ (𝜑𝜓)))
134, 5, 12cbval 1653 . 2 (∀𝑦(𝑦 ∈ {𝑥𝜑} ↔ 𝑦 ∈ {𝑥𝜓}) ↔ ∀𝑥(𝜑𝜓))
141, 13bitr2i 178 1 (∀𝑥(𝜑𝜓) ↔ {𝑥𝜑} = {𝑥𝜓})
Colors of variables: wff set class
Syntax hints:  wb 102  wal 1257   = wceq 1259  wcel 1409  [wsb 1661  {cab 2042
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-11 1413  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049
This theorem is referenced by:  abbii  2169  abbid  2170  rabbi  2504  sbcbi2  2835  dfiota2  4895  iotabi  4903  uniabio  4904
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