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Theorem iineq2 4690
Description: Equality theorem for indexed intersection. (Contributed by NM, 22-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
iineq2 (∀𝑥𝐴 𝐵 = 𝐶 𝑥𝐴 𝐵 = 𝑥𝐴 𝐶)

Proof of Theorem iineq2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eleq2 2828 . . . . 5 (𝐵 = 𝐶 → (𝑦𝐵𝑦𝐶))
21ralimi 3090 . . . 4 (∀𝑥𝐴 𝐵 = 𝐶 → ∀𝑥𝐴 (𝑦𝐵𝑦𝐶))
3 ralbi 3206 . . . 4 (∀𝑥𝐴 (𝑦𝐵𝑦𝐶) → (∀𝑥𝐴 𝑦𝐵 ↔ ∀𝑥𝐴 𝑦𝐶))
42, 3syl 17 . . 3 (∀𝑥𝐴 𝐵 = 𝐶 → (∀𝑥𝐴 𝑦𝐵 ↔ ∀𝑥𝐴 𝑦𝐶))
54abbidv 2879 . 2 (∀𝑥𝐴 𝐵 = 𝐶 → {𝑦 ∣ ∀𝑥𝐴 𝑦𝐵} = {𝑦 ∣ ∀𝑥𝐴 𝑦𝐶})
6 df-iin 4675 . 2 𝑥𝐴 𝐵 = {𝑦 ∣ ∀𝑥𝐴 𝑦𝐵}
7 df-iin 4675 . 2 𝑥𝐴 𝐶 = {𝑦 ∣ ∀𝑥𝐴 𝑦𝐶}
85, 6, 73eqtr4g 2819 1 (∀𝑥𝐴 𝐵 = 𝐶 𝑥𝐴 𝐵 = 𝑥𝐴 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1632  wcel 2139  {cab 2746  wral 3050   ciin 4673
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-ral 3055  df-iin 4675
This theorem is referenced by:  iineq2i  4692  iineq2d  4693  firest  16315  iincld  21065  elrfirn2  37779
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