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Theorem elintrabg 3939
 Description: Membership in the intersection of a class abstraction. (Contributed by NM, 17-Feb-2007.)
Assertion
Ref Expression
elintrabg (A V → (A {x B φ} ↔ x B (φA x)))
Distinct variable group:   x,A
Allowed substitution hints:   φ(x)   B(x)   V(x)

Proof of Theorem elintrabg
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 eleq1 2413 . 2 (y = A → (y {x B φ} ↔ A {x B φ}))
2 eleq1 2413 . . . 4 (y = A → (y xA x))
32imbi2d 307 . . 3 (y = A → ((φy x) ↔ (φA x)))
43ralbidv 2634 . 2 (y = A → (x B (φy x) ↔ x B (φA x)))
5 vex 2862 . . 3 y V
65elintrab 3938 . 2 (y {x B φ} ↔ x B (φy x))
71, 4, 6vtoclbg 2915 1 (A V → (A {x B φ} ↔ x B (φA x)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642   ∈ wcel 1710  ∀wral 2614  {crab 2618  ∩cint 3926 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-rab 2623  df-v 2861  df-int 3927 This theorem is referenced by: (None)
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