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

Theorem iineq2d 3833
 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 114 . . 3 (𝜑 → (𝑥𝐴𝐵 = 𝐶))
41, 3ralrimi 2503 . 2 (𝜑 → ∀𝑥𝐴 𝐵 = 𝐶)
5 iineq2 3830 . 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 103   = wceq 1331  Ⅎwnf 1436   ∈ wcel 1480  ∀wral 2416  ∩ ciin 3814 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-ral 2421  df-iin 3816 This theorem is referenced by:  iineq2dv  3835
 Copyright terms: Public domain W3C validator