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Theorem iineq12dv 45046
Description: Equality deduction for indexed intersection. (Contributed by Glauco Siliprandi, 26-Jun-2021.) Remove DV conditions. (Revised by GG, 1-Sep-2025.)
Hypotheses
Ref Expression
iineq12dv.1 (𝜑𝐴 = 𝐵)
iineq12dv.2 ((𝜑𝑥𝐵) → 𝐶 = 𝐷)
Assertion
Ref Expression
iineq12dv (𝜑 𝑥𝐴 𝐶 = 𝑥𝐵 𝐷)
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐶(𝑥)   𝐷(𝑥)

Proof of Theorem iineq12dv
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 iineq12dv.1 . . . . . . 7 (𝜑𝐴 = 𝐵)
21eleq2d 2825 . . . . . 6 (𝜑 → (𝑥𝐴𝑥𝐵))
32imbi1d 341 . . . . 5 (𝜑 → ((𝑥𝐴𝑡𝐶) ↔ (𝑥𝐵𝑡𝐶)))
43ralbidv2 3172 . . . 4 (𝜑 → (∀𝑥𝐴 𝑡𝐶 ↔ ∀𝑥𝐵 𝑡𝐶))
54abbidv 2806 . . 3 (𝜑 → {𝑡 ∣ ∀𝑥𝐴 𝑡𝐶} = {𝑡 ∣ ∀𝑥𝐵 𝑡𝐶})
6 df-iin 4999 . . 3 𝑥𝐴 𝐶 = {𝑡 ∣ ∀𝑥𝐴 𝑡𝐶}
7 df-iin 4999 . . 3 𝑥𝐵 𝐶 = {𝑡 ∣ ∀𝑥𝐵 𝑡𝐶}
85, 6, 73eqtr4g 2800 . 2 (𝜑 𝑥𝐴 𝐶 = 𝑥𝐵 𝐶)
9 iineq12dv.2 . . 3 ((𝜑𝑥𝐵) → 𝐶 = 𝐷)
109iineq2dv 5022 . 2 (𝜑 𝑥𝐵 𝐶 = 𝑥𝐵 𝐷)
118, 10eqtrd 2775 1 (𝜑 𝑥𝐴 𝐶 = 𝑥𝐵 𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  {cab 2712  wral 3059   ciin 4997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-iin 4999
This theorem is referenced by:  smflim  46733
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