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Theorem iineq12dv 39109
 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 39087 . 2 (𝜑 𝑥𝐴 𝐶 = 𝑥𝐵 𝐶)
3 iineq12dv.2 . . 3 ((𝜑𝑥𝐵) → 𝐶 = 𝐷)
43iineq2dv 4534 . 2 (𝜑 𝑥𝐵 𝐶 = 𝑥𝐵 𝐷)
52, 4eqtrd 2654 1 (𝜑 𝑥𝐴 𝐶 = 𝑥𝐵 𝐷)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   = wceq 1481   ∈ wcel 1988  ∩ ciin 4512 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-iin 4514 This theorem is referenced by:  smflim  40748
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