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Theorem iineq12dv 43028
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 43012 . 2 (𝜑 𝑥𝐴 𝐶 = 𝑥𝐵 𝐶)
3 iineq12dv.2 . . 3 ((𝜑𝑥𝐵) → 𝐶 = 𝐷)
43iineq2dv 4967 . 2 (𝜑 𝑥𝐵 𝐶 = 𝑥𝐵 𝐷)
52, 4eqtrd 2776 1 (𝜑 𝑥𝐴 𝐶 = 𝑥𝐵 𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1540  wcel 2105   ciin 4943
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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-12 2170  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3062  df-iin 4945
This theorem is referenced by:  smflim  44704
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