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Theorem intmin4 3955
 Description: Elimination of a conjunct in a class intersection. (Contributed by NM, 31-Jul-2006.)
Assertion
Ref Expression
intmin4 (A {x φ} → {x (A x φ)} = {x φ})
Distinct variable group:   x,A
Allowed substitution hint:   φ(x)

Proof of Theorem intmin4
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 ssintab 3943 . . . 4 (A {x φ} ↔ x(φA x))
2 simpr 447 . . . . . . . 8 ((A x φ) → φ)
3 ancr 532 . . . . . . . 8 ((φA x) → (φ → (A x φ)))
42, 3impbid2 195 . . . . . . 7 ((φA x) → ((A x φ) ↔ φ))
54imbi1d 308 . . . . . 6 ((φA x) → (((A x φ) → y x) ↔ (φy x)))
65alimi 1559 . . . . 5 (x(φA x) → x(((A x φ) → y x) ↔ (φy x)))
7 albi 1564 . . . . 5 (x(((A x φ) → y x) ↔ (φy x)) → (x((A x φ) → y x) ↔ x(φy x)))
86, 7syl 15 . . . 4 (x(φA x) → (x((A x φ) → y x) ↔ x(φy x)))
91, 8sylbi 187 . . 3 (A {x φ} → (x((A x φ) → y x) ↔ x(φy x)))
10 vex 2862 . . . 4 y V
1110elintab 3937 . . 3 (y {x (A x φ)} ↔ x((A x φ) → y x))
1210elintab 3937 . . 3 (y {x φ} ↔ x(φy x))
139, 11, 123bitr4g 279 . 2 (A {x φ} → (y {x (A x φ)} ↔ y {x φ}))
1413eqrdv 2351 1 (A {x φ} → {x (A x φ)} = {x φ})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540   = wceq 1642   ∈ wcel 1710  {cab 2339   ⊆ wss 3257  ∩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-nan 1288  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-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-int 3927 This theorem is referenced by: (None)
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