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Theorem iineq12f 34923
Description: Equality deduction for indexed intersections. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
Hypotheses
Ref Expression
iineq12f.1 𝑥𝐴
iineq12f.2 𝑥𝐵
Assertion
Ref Expression
iineq12f ((𝐴 = 𝐵 ∧ ∀𝑥𝐴 𝐶 = 𝐷) → 𝑥𝐴 𝐶 = 𝑥𝐵 𝐷)

Proof of Theorem iineq12f
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eleq2 2847 . . . . . 6 (𝐶 = 𝐷 → (𝑦𝐶𝑦𝐷))
21ralimi 3103 . . . . 5 (∀𝑥𝐴 𝐶 = 𝐷 → ∀𝑥𝐴 (𝑦𝐶𝑦𝐷))
3 ralbi 3110 . . . . 5 (∀𝑥𝐴 (𝑦𝐶𝑦𝐷) → (∀𝑥𝐴 𝑦𝐶 ↔ ∀𝑥𝐴 𝑦𝐷))
42, 3syl 17 . . . 4 (∀𝑥𝐴 𝐶 = 𝐷 → (∀𝑥𝐴 𝑦𝐶 ↔ ∀𝑥𝐴 𝑦𝐷))
5 iineq12f.1 . . . . 5 𝑥𝐴
6 iineq12f.2 . . . . 5 𝑥𝐵
75, 6raleqf 3330 . . . 4 (𝐴 = 𝐵 → (∀𝑥𝐴 𝑦𝐷 ↔ ∀𝑥𝐵 𝑦𝐷))
84, 7sylan9bbr 503 . . 3 ((𝐴 = 𝐵 ∧ ∀𝑥𝐴 𝐶 = 𝐷) → (∀𝑥𝐴 𝑦𝐶 ↔ ∀𝑥𝐵 𝑦𝐷))
98abbidv 2836 . 2 ((𝐴 = 𝐵 ∧ ∀𝑥𝐴 𝐶 = 𝐷) → {𝑦 ∣ ∀𝑥𝐴 𝑦𝐶} = {𝑦 ∣ ∀𝑥𝐵 𝑦𝐷})
10 df-iin 4791 . 2 𝑥𝐴 𝐶 = {𝑦 ∣ ∀𝑥𝐴 𝑦𝐶}
11 df-iin 4791 . 2 𝑥𝐵 𝐷 = {𝑦 ∣ ∀𝑥𝐵 𝑦𝐷}
129, 10, 113eqtr4g 2832 1 ((𝐴 = 𝐵 ∧ ∀𝑥𝐴 𝐶 = 𝐷) → 𝑥𝐴 𝐶 = 𝑥𝐵 𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387   = wceq 1508  wcel 2051  {cab 2751  wnfc 2909  wral 3081   ciin 4789
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-ext 2743
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ral 3086  df-iin 4791
This theorem is referenced by: (None)
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