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Theorem elintgOLD 4482
 Description: Obsolete proof of elintg 4481 as of 26-Jul-2021. (Contributed by NM, 20-Nov-2003.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
elintgOLD (𝐴𝑉 → (𝐴 𝐵 ↔ ∀𝑥𝐵 𝐴𝑥))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem elintgOLD
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eleq1 2688 . 2 (𝑦 = 𝐴 → (𝑦 𝐵𝐴 𝐵))
2 eleq1 2688 . . 3 (𝑦 = 𝐴 → (𝑦𝑥𝐴𝑥))
32ralbidv 2985 . 2 (𝑦 = 𝐴 → (∀𝑥𝐵 𝑦𝑥 ↔ ∀𝑥𝐵 𝐴𝑥))
4 vex 3201 . . 3 𝑦 ∈ V
54elint2 4480 . 2 (𝑦 𝐵 ↔ ∀𝑥𝐵 𝑦𝑥)
61, 3, 5vtoclbg 3265 1 (𝐴𝑉 → (𝐴 𝐵 ↔ ∀𝑥𝐵 𝐴𝑥))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   = wceq 1482   ∈ wcel 1989  ∀wral 2911  ∩ cint 4473 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-v 3200  df-int 4474 This theorem is referenced by: (None)
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