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Theorem bj-abbi 33613
 Description: Remove dependency on ax-13 2301 from abbi 2907. (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 2772 . 2 ({𝑥𝜑} = {𝑥𝜓} ↔ ∀𝑦(𝑦 ∈ {𝑥𝜑} ↔ 𝑦 ∈ {𝑥𝜓}))
2 bj-nfsab1 33610 . . . 4 𝑥 𝑦 ∈ {𝑥𝜑}
3 bj-nfsab1 33610 . . . 4 𝑥 𝑦 ∈ {𝑥𝜓}
42, 3nfbi 1866 . . 3 𝑥(𝑦 ∈ {𝑥𝜑} ↔ 𝑦 ∈ {𝑥𝜓})
5 nfv 1873 . . 3 𝑦(𝜑𝜓)
6 df-clab 2759 . . . . 5 (𝑦 ∈ {𝑥𝜑} ↔ [𝑦 / 𝑥]𝜑)
7 sbequ12r 2180 . . . . 5 (𝑦 = 𝑥 → ([𝑦 / 𝑥]𝜑𝜑))
86, 7syl5bb 275 . . . 4 (𝑦 = 𝑥 → (𝑦 ∈ {𝑥𝜑} ↔ 𝜑))
9 df-clab 2759 . . . . 5 (𝑦 ∈ {𝑥𝜓} ↔ [𝑦 / 𝑥]𝜓)
10 sbequ12r 2180 . . . . 5 (𝑦 = 𝑥 → ([𝑦 / 𝑥]𝜓𝜓))
119, 10syl5bb 275 . . . 4 (𝑦 = 𝑥 → (𝑦 ∈ {𝑥𝜓} ↔ 𝜓))
128, 11bibi12d 338 . . 3 (𝑦 = 𝑥 → ((𝑦 ∈ {𝑥𝜑} ↔ 𝑦 ∈ {𝑥𝜓}) ↔ (𝜑𝜓)))
134, 5, 12cbvalv1 2277 . 2 (∀𝑦(𝑦 ∈ {𝑥𝜑} ↔ 𝑦 ∈ {𝑥𝜓}) ↔ ∀𝑥(𝜑𝜓))
141, 13bitr2i 268 1 (∀𝑥(𝜑𝜓) ↔ {𝑥𝜑} = {𝑥𝜓})
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 198  ∀wal 1505   = wceq 1507  [wsb 2015   ∈ wcel 2050  {cab 2758 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2750 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2759  df-cleq 2771 This theorem is referenced by:  bj-abbid  33615
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