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Theorem eliminable-abelab 34327
 Description: A theorem used to prove the base case of the Eliminability Theorem (see section comment): abstraction belongs to abstraction. (Contributed by BJ, 30-Apr-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
eliminable-abelab ({𝑥𝜑} ∈ {𝑦𝜓} ↔ ∃𝑧(∀𝑡(𝑡𝑧 ↔ [𝑡 / 𝑥]𝜑) ∧ [𝑧 / 𝑦]𝜓))
Distinct variable groups:   𝑥,𝑡,𝑧   𝑦,𝑧   𝜑,𝑧,𝑡   𝜓,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦,𝑡)

Proof of Theorem eliminable-abelab
StepHypRef Expression
1 dfclel 2871 . 2 ({𝑥𝜑} ∈ {𝑦𝜓} ↔ ∃𝑧(𝑧 = {𝑥𝜑} ∧ 𝑧 ∈ {𝑦𝜓}))
2 eliminable-veqab 34323 . . . 4 (𝑧 = {𝑥𝜑} ↔ ∀𝑡(𝑡𝑧 ↔ [𝑡 / 𝑥]𝜑))
3 eliminable-velab 34322 . . . 4 (𝑧 ∈ {𝑦𝜓} ↔ [𝑧 / 𝑦]𝜓)
42, 3anbi12i 629 . . 3 ((𝑧 = {𝑥𝜑} ∧ 𝑧 ∈ {𝑦𝜓}) ↔ (∀𝑡(𝑡𝑧 ↔ [𝑡 / 𝑥]𝜑) ∧ [𝑧 / 𝑦]𝜓))
54exbii 1849 . 2 (∃𝑧(𝑧 = {𝑥𝜑} ∧ 𝑧 ∈ {𝑦𝜓}) ↔ ∃𝑧(∀𝑡(𝑡𝑧 ↔ [𝑡 / 𝑥]𝜑) ∧ [𝑧 / 𝑦]𝜓))
61, 5bitri 278 1 ({𝑥𝜑} ∈ {𝑦𝜓} ↔ ∃𝑧(∀𝑡(𝑡𝑧 ↔ [𝑡 / 𝑥]𝜑) ∧ [𝑧 / 𝑦]𝜓))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399  ∀wal 1536   = wceq 1538  ∃wex 1781  [wsb 2069   ∈ wcel 2111  {cab 2776 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-clab 2777  df-cleq 2791  df-clel 2870 This theorem is referenced by: (None)
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