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 ∈ {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 2758	. 2 ⊢ (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ [𝑥 / 𝑦]𝜑)
2		ax6ev 2059	. . . 4 ⊢ ∃𝑧 𝑧 = 𝑥
3	2	biantrur 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))
5	4	bicomi 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))
7	6	equcoms 2105	. . . . 5 ⊢ (𝑧 = 𝑥 → ([𝑥 / 𝑦]𝜑 ↔ [𝑧 / 𝑦]𝜑))
8	7	pm5.32i 564	. . . 4 ⊢ ((𝑧 = 𝑥 ∧ [𝑥 / 𝑦]𝜑) ↔ (𝑧 = 𝑥 ∧ [𝑧 / 𝑦]𝜑))
9	8	exbii 1924	. . 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 2758	. . . . 5 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜑} ↔ [𝑧 / 𝑦]𝜑)
12	11	bicomi 214	. . . 4 ⊢ ([𝑧 / 𝑦]𝜑 ↔ 𝑧 ∈ {𝑦 ∣ 𝜑})
13	12	anbi2i 609	. . 3 ⊢ ((𝑧 = 𝑥 ∧ [𝑧 / 𝑦]𝜑) ↔ (𝑧 = 𝑥 ∧ 𝑧 ∈ {𝑦 ∣ 𝜑}))
14	13	exbii 1924	. 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 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)