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Theorem iineq2d 4729
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 402 . . 3 (𝜑 → (𝑥𝐴𝐵 = 𝐶))
41, 3ralrimi 3136 . 2 (𝜑 → ∀𝑥𝐴 𝐵 = 𝐶)
5 iineq2 4726 . 2 (∀𝑥𝐴 𝐵 = 𝐶 𝑥𝐴 𝐵 = 𝑥𝐴 𝐶)
64, 5syl 17 1 (𝜑 𝑥𝐴 𝐵 = 𝑥𝐴 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wnf 1879  wcel 2157  wral 3087   ciin 4709
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2775
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2784  df-cleq 2790  df-clel 2793  df-ral 3092  df-iin 4711
This theorem is referenced by:  iineq2dv  4731  pmapglbx  35782  smflimmpt  41750  smfsupmpt  41755  smfinfmpt  41759  smflimsuplem4  41763  smflimsupmpt  41769  smfliminfmpt  41772
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