Theorem elintabOLD 4966

Description: Obsolete version of elintab 4965 as of 17-Jan-2025. (Contributed by NM, 30-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
elintab.ex	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
elintabOLD	⊢ (𝐴 ∈ ∩ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝜑 → 𝐴 ∈ 𝑥))

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem elintabOLD

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elintab.ex	. . 3 ⊢ 𝐴 ∈ V
2	1	elint 4959	. 2 ⊢ (𝐴 ∈ ∩ {𝑥 ∣ 𝜑} ↔ ∀𝑦(𝑦 ∈ {𝑥 ∣ 𝜑} → 𝐴 ∈ 𝑦))
3		nfsab1 2713	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∣ 𝜑}
4		nfv 1909	. . . 4 ⊢ Ⅎ𝑥 𝐴 ∈ 𝑦
5	3, 4	nfim 1891	. . 3 ⊢ Ⅎ𝑥(𝑦 ∈ {𝑥 ∣ 𝜑} → 𝐴 ∈ 𝑦)
6		nfv 1909	. . 3 ⊢ Ⅎ𝑦(𝜑 → 𝐴 ∈ 𝑥)
7		eleq1w 2812	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ {𝑥 ∣ 𝜑}))
8		abid 2709	. . . . 5 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
9	7, 8	bitrdi 286	. . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑))
10		eleq2w 2813	. . . 4 ⊢ (𝑦 = 𝑥 → (𝐴 ∈ 𝑦 ↔ 𝐴 ∈ 𝑥))
11	9, 10	imbi12d 343	. . 3 ⊢ (𝑦 = 𝑥 → ((𝑦 ∈ {𝑥 ∣ 𝜑} → 𝐴 ∈ 𝑦) ↔ (𝜑 → 𝐴 ∈ 𝑥)))
12	5, 6, 11	cbvalv1 2332	. 2 ⊢ (∀𝑦(𝑦 ∈ {𝑥 ∣ 𝜑} → 𝐴 ∈ 𝑦) ↔ ∀𝑥(𝜑 → 𝐴 ∈ 𝑥))
13	2, 12	bitri 274	1 ⊢ (𝐴 ∈ ∩ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝜑 → 𝐴 ∈ 𝑥))

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1531   ∈ wcel 2098  {cab 2705  Vcvv 3473  ∩ cint 4953

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-int 4954

This theorem is referenced by: (None)