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Theorem iineq12i 36139
Description: Equality theorem for indexed intersection. Inference version. General version of iineq1i 36138. (Contributed by GG, 1-Sep-2025.)
Hypotheses
Ref Expression
iineq12i.1 𝐴 = 𝐵
iineq12i.2 𝐶 = 𝐷
Assertion
Ref Expression
iineq12i 𝑥𝐴 𝐶 = 𝑥𝐵 𝐷

Proof of Theorem iineq12i
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 iineq12i.1 . . . 4 𝐴 = 𝐵
2 iineq12i.2 . . . . 5 𝐶 = 𝐷
32eleq2i 2829 . . . 4 (𝑡𝐶𝑡𝐷)
41, 3raleqbii 3340 . . 3 (∀𝑥𝐴 𝑡𝐶 ↔ ∀𝑥𝐵 𝑡𝐷)
54abbii 2805 . 2 {𝑡 ∣ ∀𝑥𝐴 𝑡𝐶} = {𝑡 ∣ ∀𝑥𝐵 𝑡𝐷}
6 df-iin 5002 . 2 𝑥𝐴 𝐶 = {𝑡 ∣ ∀𝑥𝐴 𝑡𝐶}
7 df-iin 5002 . 2 𝑥𝐵 𝐷 = {𝑡 ∣ ∀𝑥𝐵 𝑡𝐷}
85, 6, 73eqtr4i 2771 1 𝑥𝐴 𝐶 = 𝑥𝐵 𝐷
Colors of variables: wff setvar class
Syntax hints:   = wceq 1535  wcel 2104  {cab 2710  wral 3057   ciin 5000
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-ext 2704
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1775  df-sb 2061  df-clab 2711  df-cleq 2725  df-clel 2812  df-ral 3058  df-iin 5002
This theorem is referenced by: (None)
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