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Theorem iineq12f 35916
 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 2840 . . . . . 6 (𝐶 = 𝐷 → (𝑦𝐶𝑦𝐷))
21ralimi 3092 . . . . 5 (∀𝑥𝐴 𝐶 = 𝐷 → ∀𝑥𝐴 (𝑦𝐶𝑦𝐷))
3 ralbi 3099 . . . . 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 3315 . . . 4 (𝐴 = 𝐵 → (∀𝑥𝐴 𝑦𝐷 ↔ ∀𝑥𝐵 𝑦𝐷))
84, 7sylan9bbr 514 . . 3 ((𝐴 = 𝐵 ∧ ∀𝑥𝐴 𝐶 = 𝐷) → (∀𝑥𝐴 𝑦𝐶 ↔ ∀𝑥𝐵 𝑦𝐷))
98abbidv 2822 . 2 ((𝐴 = 𝐵 ∧ ∀𝑥𝐴 𝐶 = 𝐷) → {𝑦 ∣ ∀𝑥𝐴 𝑦𝐶} = {𝑦 ∣ ∀𝑥𝐵 𝑦𝐷})
10 df-iin 4889 . 2 𝑥𝐴 𝐶 = {𝑦 ∣ ∀𝑥𝐴 𝑦𝐶}
11 df-iin 4889 . 2 𝑥𝐵 𝐷 = {𝑦 ∣ ∀𝑥𝐵 𝑦𝐷}
129, 10, 113eqtr4g 2818 1 ((𝐴 = 𝐵 ∧ ∀𝑥𝐴 𝐶 = 𝐷) → 𝑥𝐴 𝐶 = 𝑥𝐵 𝐷)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {cab 2735  Ⅎwnfc 2899  ∀wral 3070  ∩ ciin 4887 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-iin 4889 This theorem is referenced by: (None)
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