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Theorem bj-cleljustab 33181
Description: An instance of df-clel 2767 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 2758 (hence without df-clel 2767 or df-cleq 2764) was stressed by Mario Carneiro. The instance of df-clel 2767 where the LHS has the form "setvar setvar" is proved as cleljust 2153, from FOL= and ax-8 2147. Note: when df-ssb 32958 is the official definition for substitution, one can use bj-ssbequ 32967 instead of sbequ 2523 to prove bj-cleljustab 33181 from Tarski's FOL= with df-clab 2758. (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 2758 . 2 (𝑥 ∈ {𝑦𝜑} ↔ [𝑥 / 𝑦]𝜑)
2 ax6ev 2059 . . . 4 𝑧 𝑧 = 𝑥
32biantrur 520 . . 3 ([𝑥 / 𝑦]𝜑 ↔ (∃𝑧 𝑧 = 𝑥 ∧ [𝑥 / 𝑦]𝜑))
4 19.41v 2029 . . . 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 2523 . . . . . 6 (𝑥 = 𝑧 → ([𝑥 / 𝑦]𝜑 ↔ [𝑧 / 𝑦]𝜑))
76equcoms 2105 . . . . 5 (𝑧 = 𝑥 → ([𝑥 / 𝑦]𝜑 ↔ [𝑧 / 𝑦]𝜑))
87pm5.32i 564 . . . 4 ((𝑧 = 𝑥 ∧ [𝑥 / 𝑦]𝜑) ↔ (𝑧 = 𝑥 ∧ [𝑧 / 𝑦]𝜑))
98exbii 1924 . . 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 2758 . . . . 5 (𝑧 ∈ {𝑦𝜑} ↔ [𝑧 / 𝑦]𝜑)
1211bicomi 214 . . . 4 ([𝑧 / 𝑦]𝜑𝑧 ∈ {𝑦𝜑})
1312anbi2i 609 . . 3 ((𝑧 = 𝑥 ∧ [𝑧 / 𝑦]𝜑) ↔ (𝑧 = 𝑥𝑧 ∈ {𝑦𝜑}))
1413exbii 1924 . 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 382  wex 1852  [wsb 2049  wcel 2145  {cab 2757
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-10 2174  ax-12 2203  ax-13 2408
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758
This theorem is referenced by: (None)
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