Theorem clelabOLD 2896

Description: Obsolete version of clelab 2895 as of 2-Sep-2024. (Contributed by NM, 23-Dec-1993.) (Proof shortened by Wolf Lammen, 16-Nov-2019.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
clelabOLD	⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑))

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem clelabOLD

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfclel 2831	. 2 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑦(𝑦 = 𝐴 ∧ 𝑦 ∈ {𝑥 ∣ 𝜑}))
2		nfv 1915	. . 3 ⊢ Ⅎ𝑦(𝑥 = 𝐴 ∧ 𝜑)
3		nfv 1915	. . . 4 ⊢ Ⅎ𝑥 𝑦 = 𝐴
4		nfsab1 2744	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∣ 𝜑}
5	3, 4	nfan 1900	. . 3 ⊢ Ⅎ𝑥(𝑦 = 𝐴 ∧ 𝑦 ∈ {𝑥 ∣ 𝜑})
6		eqeq1 2762	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝐴 ↔ 𝑦 = 𝐴))
7		sbequ12 2250	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
8		df-clab 2736	. . . . 5 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑)
9	7, 8	bitr4di 292	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝑦 ∈ {𝑥 ∣ 𝜑}))
10	6, 9	anbi12d 633	. . 3 ⊢ (𝑥 = 𝑦 → ((𝑥 = 𝐴 ∧ 𝜑) ↔ (𝑦 = 𝐴 ∧ 𝑦 ∈ {𝑥 ∣ 𝜑})))
11	2, 5, 10	cbvexv1 2351	. 2 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ ∃𝑦(𝑦 = 𝐴 ∧ 𝑦 ∈ {𝑥 ∣ 𝜑}))
12	1, 11	bitr4i 281	1 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑))

Syntax hints:   ↔ wb 209   ∧ wa 399   = wceq 1538  ∃wex 1781  [wsb 2069   ∈ wcel 2111  {cab 2735

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830

This theorem is referenced by: (None)