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Theorem iinconst 4501
Description: Indexed intersection of a constant class, i.e. where 𝐵 does not depend on 𝑥. (Contributed by Mario Carneiro, 6-Feb-2015.)
Assertion
Ref Expression
iinconst (𝐴 ≠ ∅ → 𝑥𝐴 𝐵 = 𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem iinconst
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 r19.3rzv 4041 . . 3 (𝐴 ≠ ∅ → (𝑦𝐵 ↔ ∀𝑥𝐴 𝑦𝐵))
2 vex 3192 . . . 4 𝑦 ∈ V
3 eliin 4496 . . . 4 (𝑦 ∈ V → (𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑦𝐵))
42, 3ax-mp 5 . . 3 (𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑦𝐵)
51, 4syl6rbbr 279 . 2 (𝐴 ≠ ∅ → (𝑦 𝑥𝐴 𝐵𝑦𝐵))
65eqrdv 2619 1 (𝐴 ≠ ∅ → 𝑥𝐴 𝐵 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1480  wcel 1987  wne 2790  wral 2907  Vcvv 3189  c0 3896   ciin 4491
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-v 3191  df-dif 3562  df-nul 3897  df-iin 4493
This theorem is referenced by:  iin0  4804  ptbasfi  21307
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