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Theorem bj-cleljustab 32822
Description: An instance of df-clel 2616 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 2607 (hence without df-clel 2616 or df-cleq 2613) was stressed by Mario Carneiro. The instance of df-clel 2616 where the LHS has the form "setvar setvar" is proved as cleljust 1996, from FOL= and ax-8 1990. Note: when df-ssb 32595 is the official definition for substitution, one can use bj-ssbequ instead of sbequ 2374 to prove bj-cleljustab 32822 from Tarski's FOL= with df-clab 2607. (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 2607 . 2 (𝑥 ∈ {𝑦𝜑} ↔ [𝑥 / 𝑦]𝜑)
2 ax6ev 1888 . . . 4 𝑧 𝑧 = 𝑥
32biantrur 527 . . 3 ([𝑥 / 𝑦]𝜑 ↔ (∃𝑧 𝑧 = 𝑥 ∧ [𝑥 / 𝑦]𝜑))
4 19.41v 1912 . . . 4 (∃𝑧(𝑧 = 𝑥 ∧ [𝑥 / 𝑦]𝜑) ↔ (∃𝑧 𝑧 = 𝑥 ∧ [𝑥 / 𝑦]𝜑))
54bicomi 214 . . 3 ((∃𝑧 𝑧 = 𝑥 ∧ [𝑥 / 𝑦]𝜑) ↔ ∃𝑧(𝑧 = 𝑥 ∧ [𝑥 / 𝑦]𝜑))
6 sbequ 2374 . . . . . 6 (𝑥 = 𝑧 → ([𝑥 / 𝑦]𝜑 ↔ [𝑧 / 𝑦]𝜑))
76equcoms 1945 . . . . 5 (𝑧 = 𝑥 → ([𝑥 / 𝑦]𝜑 ↔ [𝑧 / 𝑦]𝜑))
87pm5.32i 668 . . . 4 ((𝑧 = 𝑥 ∧ [𝑥 / 𝑦]𝜑) ↔ (𝑧 = 𝑥 ∧ [𝑧 / 𝑦]𝜑))
98exbii 1772 . . 3 (∃𝑧(𝑧 = 𝑥 ∧ [𝑥 / 𝑦]𝜑) ↔ ∃𝑧(𝑧 = 𝑥 ∧ [𝑧 / 𝑦]𝜑))
103, 5, 93bitri 286 . 2 ([𝑥 / 𝑦]𝜑 ↔ ∃𝑧(𝑧 = 𝑥 ∧ [𝑧 / 𝑦]𝜑))
11 df-clab 2607 . . . . 5 (𝑧 ∈ {𝑦𝜑} ↔ [𝑧 / 𝑦]𝜑)
1211bicomi 214 . . . 4 ([𝑧 / 𝑦]𝜑𝑧 ∈ {𝑦𝜑})
1312anbi2i 729 . . 3 ((𝑧 = 𝑥 ∧ [𝑧 / 𝑦]𝜑) ↔ (𝑧 = 𝑥𝑧 ∈ {𝑦𝜑}))
1413exbii 1772 . 2 (∃𝑧(𝑧 = 𝑥 ∧ [𝑧 / 𝑦]𝜑) ↔ ∃𝑧(𝑧 = 𝑥𝑧 ∈ {𝑦𝜑}))
151, 10, 143bitri 286 1 (𝑥 ∈ {𝑦𝜑} ↔ ∃𝑧(𝑧 = 𝑥𝑧 ∈ {𝑦𝜑}))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384  wex 1702  [wsb 1878  wcel 1988  {cab 2606
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-10 2017  ax-12 2045  ax-13 2244
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607
This theorem is referenced by: (None)
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