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Theorem bj-cleljustab 33422
 Description: An instance of df-clel 2773 where the LHS (the definiendum) has the form "setvar ∈ class abstraction". The straightforward yet important fact that this statement can be proved from FOL= and df-clab 2763 (hence without df-clel 2773 or df-cleq 2769) was stressed by Mario Carneiro. The instance of df-clel 2773 where the LHS has the form "setvar ∈ setvar" is proved as cleljust 2114, from FOL= and ax-8 2108. Note: when df-ssb 33211 is the official definition for substitution, one can use bj-ssbequ 33220 instead of sbequ 2451 to prove bj-cleljustab 33422 from Tarski's FOL= with df-clab 2763. (Contributed by BJ, 8-Nov-2021.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-cleljustab (𝑥 ∈ {𝑦𝜑} ↔ ∃𝑧(𝑧 = 𝑥𝑧 ∈ {𝑦𝜑}))
Distinct variable groups:   𝑥,𝑧   𝑦,𝑧   𝜑,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem bj-cleljustab
StepHypRef Expression
1 df-clab 2763 . 2 (𝑥 ∈ {𝑦𝜑} ↔ [𝑥 / 𝑦]𝜑)
2 ax6ev 2023 . . . 4 𝑧 𝑧 = 𝑥
32biantrur 526 . . 3 ([𝑥 / 𝑦]𝜑 ↔ (∃𝑧 𝑧 = 𝑥 ∧ [𝑥 / 𝑦]𝜑))
4 19.41v 1992 . . . 4 (∃𝑧(𝑧 = 𝑥 ∧ [𝑥 / 𝑦]𝜑) ↔ (∃𝑧 𝑧 = 𝑥 ∧ [𝑥 / 𝑦]𝜑))
54bicomi 216 . . 3 ((∃𝑧 𝑧 = 𝑥 ∧ [𝑥 / 𝑦]𝜑) ↔ ∃𝑧(𝑧 = 𝑥 ∧ [𝑥 / 𝑦]𝜑))
6 sbequ 2451 . . . . . 6 (𝑥 = 𝑧 → ([𝑥 / 𝑦]𝜑 ↔ [𝑧 / 𝑦]𝜑))
76equcoms 2066 . . . . 5 (𝑧 = 𝑥 → ([𝑥 / 𝑦]𝜑 ↔ [𝑧 / 𝑦]𝜑))
87pm5.32i 570 . . . 4 ((𝑧 = 𝑥 ∧ [𝑥 / 𝑦]𝜑) ↔ (𝑧 = 𝑥 ∧ [𝑧 / 𝑦]𝜑))
98exbii 1892 . . 3 (∃𝑧(𝑧 = 𝑥 ∧ [𝑥 / 𝑦]𝜑) ↔ ∃𝑧(𝑧 = 𝑥 ∧ [𝑧 / 𝑦]𝜑))
103, 5, 93bitri 289 . 2 ([𝑥 / 𝑦]𝜑 ↔ ∃𝑧(𝑧 = 𝑥 ∧ [𝑧 / 𝑦]𝜑))
11 df-clab 2763 . . . . 5 (𝑧 ∈ {𝑦𝜑} ↔ [𝑧 / 𝑦]𝜑)
1211bicomi 216 . . . 4 ([𝑧 / 𝑦]𝜑𝑧 ∈ {𝑦𝜑})
1312anbi2i 616 . . 3 ((𝑧 = 𝑥 ∧ [𝑧 / 𝑦]𝜑) ↔ (𝑧 = 𝑥𝑧 ∈ {𝑦𝜑}))
1413exbii 1892 . 2 (∃𝑧(𝑧 = 𝑥 ∧ [𝑧 / 𝑦]𝜑) ↔ ∃𝑧(𝑧 = 𝑥𝑧 ∈ {𝑦𝜑}))
151, 10, 143bitri 289 1 (𝑥 ∈ {𝑦𝜑} ↔ ∃𝑧(𝑧 = 𝑥𝑧 ∈ {𝑦𝜑}))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 198   ∧ wa 386  ∃wex 1823  [wsb 2011   ∈ wcel 2106  {cab 2762 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-10 2134  ax-12 2162  ax-13 2333 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2763 This theorem is referenced by: (None)
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