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Theorem iineq2 3901
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 2241 . . . . 5 (𝐵 = 𝐶 → (𝑦𝐵𝑦𝐶))
21ralimi 2540 . . . 4 (∀𝑥𝐴 𝐵 = 𝐶 → ∀𝑥𝐴 (𝑦𝐵𝑦𝐶))
3 ralbi 2609 . . . 4 (∀𝑥𝐴 (𝑦𝐵𝑦𝐶) → (∀𝑥𝐴 𝑦𝐵 ↔ ∀𝑥𝐴 𝑦𝐶))
42, 3syl 14 . . 3 (∀𝑥𝐴 𝐵 = 𝐶 → (∀𝑥𝐴 𝑦𝐵 ↔ ∀𝑥𝐴 𝑦𝐶))
54abbidv 2295 . 2 (∀𝑥𝐴 𝐵 = 𝐶 → {𝑦 ∣ ∀𝑥𝐴 𝑦𝐵} = {𝑦 ∣ ∀𝑥𝐴 𝑦𝐶})
6 df-iin 3887 . 2 𝑥𝐴 𝐵 = {𝑦 ∣ ∀𝑥𝐴 𝑦𝐵}
7 df-iin 3887 . 2 𝑥𝐴 𝐶 = {𝑦 ∣ ∀𝑥𝐴 𝑦𝐶}
85, 6, 73eqtr4g 2235 1 (∀𝑥𝐴 𝐵 = 𝐶 𝑥𝐴 𝐵 = 𝑥𝐴 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105   = wceq 1353  wcel 2148  {cab 2163  wral 2455   ciin 3885
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-ral 2460  df-iin 3887
This theorem is referenced by:  iineq2i  3903  iineq2d  3904
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