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Theorem iineq2d 4914
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 415 . . 3 (𝜑 → (𝑥𝐴𝐵 = 𝐶))
41, 3ralrimi 3203 . 2 (𝜑 → ∀𝑥𝐴 𝐵 = 𝐶)
5 iineq2 4911 . 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 398   = wceq 1537  wnf 1784  wcel 2114  wral 3125   ciin 4892
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-12 2177  ax-ext 2792
This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2799  df-cleq 2813  df-clel 2891  df-ral 3130  df-iin 4894
This theorem is referenced by:  iineq2dv  4916  pmapglbx  36937  smflimmpt  43228  smfsupmpt  43233  smfinfmpt  43237  smflimsuplem4  43241  smflimsupmpt  43247  smfliminfmpt  43250
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