ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  iineq2d GIF version

Theorem iineq2d 3727
Description: Equality deduction for indexed intersection. (Contributed by NM, 7-Dec-2011.)
Hypotheses
Ref Expression
iineq2d.1 𝑥𝜑
iineq2d.2 ((𝜑𝑥𝐴) → 𝐵 = 𝐶)
Assertion
Ref Expression
iineq2d (𝜑 𝑥𝐴 𝐵 = 𝑥𝐴 𝐶)

Proof of Theorem iineq2d
StepHypRef Expression
1 iineq2d.1 . . 3 𝑥𝜑
2 iineq2d.2 . . . 4 ((𝜑𝑥𝐴) → 𝐵 = 𝐶)
32ex 113 . . 3 (𝜑 → (𝑥𝐴𝐵 = 𝐶))
41, 3ralrimi 2440 . 2 (𝜑 → ∀𝑥𝐴 𝐵 = 𝐶)
5 iineq2 3724 . 2 (∀𝑥𝐴 𝐵 = 𝐶 𝑥𝐴 𝐵 = 𝑥𝐴 𝐶)
64, 5syl 14 1 (𝜑 𝑥𝐴 𝐵 = 𝑥𝐴 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1287  wnf 1392  wcel 1436  wral 2355   ciin 3708
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-11 1440  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-ral 2360  df-iin 3710
This theorem is referenced by:  iineq2dv  3729
  Copyright terms: Public domain W3C validator