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Theorem bj-abbi 33052
Description: Remove dependency on ax-13 2379 from abbi 2863. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-abbi (∀𝑥(𝜑𝜓) ↔ {𝑥𝜑} = {𝑥𝜓})

Proof of Theorem bj-abbi
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 dfcleq 2742 . 2 ({𝑥𝜑} = {𝑥𝜓} ↔ ∀𝑦(𝑦 ∈ {𝑥𝜑} ↔ 𝑦 ∈ {𝑥𝜓}))
2 bj-nfsab1 33049 . . . 4 𝑥 𝑦 ∈ {𝑥𝜑}
3 bj-nfsab1 33049 . . . 4 𝑥 𝑦 ∈ {𝑥𝜓}
42, 3nfbi 1970 . . 3 𝑥(𝑦 ∈ {𝑥𝜑} ↔ 𝑦 ∈ {𝑥𝜓})
5 nfv 1980 . . 3 𝑦(𝜑𝜓)
6 df-clab 2735 . . . . 5 (𝑦 ∈ {𝑥𝜑} ↔ [𝑦 / 𝑥]𝜑)
7 sbequ12r 2247 . . . . 5 (𝑦 = 𝑥 → ([𝑦 / 𝑥]𝜑𝜑))
86, 7syl5bb 272 . . . 4 (𝑦 = 𝑥 → (𝑦 ∈ {𝑥𝜑} ↔ 𝜑))
9 df-clab 2735 . . . . 5 (𝑦 ∈ {𝑥𝜓} ↔ [𝑦 / 𝑥]𝜓)
10 sbequ12r 2247 . . . . 5 (𝑦 = 𝑥 → ([𝑦 / 𝑥]𝜓𝜓))
119, 10syl5bb 272 . . . 4 (𝑦 = 𝑥 → (𝑦 ∈ {𝑥𝜓} ↔ 𝜓))
128, 11bibi12d 334 . . 3 (𝑦 = 𝑥 → ((𝑦 ∈ {𝑥𝜑} ↔ 𝑦 ∈ {𝑥𝜓}) ↔ (𝜑𝜓)))
134, 5, 12cbvalv1 2308 . 2 (∀𝑦(𝑦 ∈ {𝑥𝜑} ↔ 𝑦 ∈ {𝑥𝜓}) ↔ ∀𝑥(𝜑𝜓))
141, 13bitr2i 265 1 (∀𝑥(𝜑𝜓) ↔ {𝑥𝜑} = {𝑥𝜓})
Colors of variables: wff setvar class
Syntax hints:  wb 196  wal 1618   = wceq 1620  [wsb 2034  wcel 2127  {cab 2734
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-ext 2728
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-clab 2735  df-cleq 2741
This theorem is referenced by:  bj-abbii  33054  bj-abbid  33055
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