Theorem bj-cleljustab 32847

Description: An instance of df-clel 2618 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 2609 (hence without df-clel 2618 or df-cleq 2615) was stressed by Mario Carneiro. The instance of df-clel 2618 where the LHS has the form "setvar ∈ setvar" is proved as cleljust 1998, from FOL= and ax-8 1992. Note: when df-ssb 32620 is the official definition for substitution, one can use bj-ssbequ instead of sbequ 2376 to prove bj-cleljustab 32847 from Tarski's FOL= with df-clab 2609. (Contributed by BJ, 8-Nov-2021.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-cleljustab	⊢ (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑧 ∈ {𝑦 ∣ 𝜑}))

Distinct variable groups:   𝑥,𝑧   𝑦,𝑧   𝜑,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem bj-cleljustab

Step	Hyp	Ref	Expression
1		df-clab 2609	. 2 ⊢ (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ [𝑥 / 𝑦]𝜑)
2		ax6ev 1890	. . . 4 ⊢ ∃𝑧 𝑧 = 𝑥
3	2	biantrur 527	. . 3 ⊢ ([𝑥 / 𝑦]𝜑 ↔ (∃𝑧 𝑧 = 𝑥 ∧ [𝑥 / 𝑦]𝜑))
4		19.41v 1914	. . . 4 ⊢ (∃𝑧(𝑧 = 𝑥 ∧ [𝑥 / 𝑦]𝜑) ↔ (∃𝑧 𝑧 = 𝑥 ∧ [𝑥 / 𝑦]𝜑))
5	4	bicomi 214	. . 3 ⊢ ((∃𝑧 𝑧 = 𝑥 ∧ [𝑥 / 𝑦]𝜑) ↔ ∃𝑧(𝑧 = 𝑥 ∧ [𝑥 / 𝑦]𝜑))
6		sbequ 2376	. . . . . 6 ⊢ (𝑥 = 𝑧 → ([𝑥 / 𝑦]𝜑 ↔ [𝑧 / 𝑦]𝜑))
7	6	equcoms 1947	. . . . 5 ⊢ (𝑧 = 𝑥 → ([𝑥 / 𝑦]𝜑 ↔ [𝑧 / 𝑦]𝜑))
8	7	pm5.32i 669	. . . 4 ⊢ ((𝑧 = 𝑥 ∧ [𝑥 / 𝑦]𝜑) ↔ (𝑧 = 𝑥 ∧ [𝑧 / 𝑦]𝜑))
9	8	exbii 1774	. . 3 ⊢ (∃𝑧(𝑧 = 𝑥 ∧ [𝑥 / 𝑦]𝜑) ↔ ∃𝑧(𝑧 = 𝑥 ∧ [𝑧 / 𝑦]𝜑))
10	3, 5, 9	3bitri 286	. 2 ⊢ ([𝑥 / 𝑦]𝜑 ↔ ∃𝑧(𝑧 = 𝑥 ∧ [𝑧 / 𝑦]𝜑))
11		df-clab 2609	. . . . 5 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜑} ↔ [𝑧 / 𝑦]𝜑)
12	11	bicomi 214	. . . 4 ⊢ ([𝑧 / 𝑦]𝜑 ↔ 𝑧 ∈ {𝑦 ∣ 𝜑})
13	12	anbi2i 730	. . 3 ⊢ ((𝑧 = 𝑥 ∧ [𝑧 / 𝑦]𝜑) ↔ (𝑧 = 𝑥 ∧ 𝑧 ∈ {𝑦 ∣ 𝜑}))
14	13	exbii 1774	. 2 ⊢ (∃𝑧(𝑧 = 𝑥 ∧ [𝑧 / 𝑦]𝜑) ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑧 ∈ {𝑦 ∣ 𝜑}))
15	1, 10, 14	3bitri 286	1 ⊢ (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑧 ∈ {𝑦 ∣ 𝜑}))

Syntax hints:   ↔ wb 196   ∧ wa 384  ∃wex 1704  [wsb 1880   ∈ wcel 1990  {cab 2608

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609

This theorem is referenced by: (None)