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Theorem iineq12dv 42108
 Description: Equality deduction for indexed intersection. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypotheses
Ref Expression
iineq12dv.1 (𝜑𝐴 = 𝐵)
iineq12dv.2 ((𝜑𝑥𝐵) → 𝐶 = 𝐷)
Assertion
Ref Expression
iineq12dv (𝜑 𝑥𝐴 𝐶 = 𝑥𝐵 𝐷)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜑,𝑥
Allowed substitution hints:   𝐶(𝑥)   𝐷(𝑥)

Proof of Theorem iineq12dv
StepHypRef Expression
1 iineq12dv.1 . . 3 (𝜑𝐴 = 𝐵)
21iineq1d 42092 . 2 (𝜑 𝑥𝐴 𝐶 = 𝑥𝐵 𝐶)
3 iineq12dv.2 . . 3 ((𝜑𝑥𝐵) → 𝐶 = 𝐷)
43iineq2dv 4909 . 2 (𝜑 𝑥𝐵 𝐶 = 𝑥𝐵 𝐷)
52, 4eqtrd 2794 1 (𝜑 𝑥𝐴 𝐶 = 𝑥𝐵 𝐷)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 400   = wceq 1539   ∈ wcel 2112  ∩ ciin 4885 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-12 2176  ax-ext 2730 This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1783  df-nf 1787  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-ral 3076  df-iin 4887 This theorem is referenced by:  smflim  43769
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