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Theorem elintabOLD 4964
Description: Obsolete version of elintab 4963 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 {𝑥𝜑} ↔ ∀𝑥(𝜑𝐴𝑥))
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem elintabOLD
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 elintab.ex . . 3 𝐴 ∈ V
21elint 4957 . 2 (𝐴 {𝑥𝜑} ↔ ∀𝑦(𝑦 ∈ {𝑥𝜑} → 𝐴𝑦))
3 nfsab1 2720 . . . 4 𝑥 𝑦 ∈ {𝑥𝜑}
4 nfv 1912 . . . 4 𝑥 𝐴𝑦
53, 4nfim 1894 . . 3 𝑥(𝑦 ∈ {𝑥𝜑} → 𝐴𝑦)
6 nfv 1912 . . 3 𝑦(𝜑𝐴𝑥)
7 eleq1w 2822 . . . . 5 (𝑦 = 𝑥 → (𝑦 ∈ {𝑥𝜑} ↔ 𝑥 ∈ {𝑥𝜑}))
8 abid 2716 . . . . 5 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
97, 8bitrdi 287 . . . 4 (𝑦 = 𝑥 → (𝑦 ∈ {𝑥𝜑} ↔ 𝜑))
10 eleq2w 2823 . . . 4 (𝑦 = 𝑥 → (𝐴𝑦𝐴𝑥))
119, 10imbi12d 344 . . 3 (𝑦 = 𝑥 → ((𝑦 ∈ {𝑥𝜑} → 𝐴𝑦) ↔ (𝜑𝐴𝑥)))
125, 6, 11cbvalv1 2342 . 2 (∀𝑦(𝑦 ∈ {𝑥𝜑} → 𝐴𝑦) ↔ ∀𝑥(𝜑𝐴𝑥))
132, 12bitri 275 1 (𝐴 {𝑥𝜑} ↔ ∀𝑥(𝜑𝐴𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wal 1535  wcel 2106  {cab 2712  Vcvv 3478   cint 4951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-int 4952
This theorem is referenced by: (None)
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