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Theorem fveleq 31484
Description: Please add description here. (Contributed by Jeff Hoffman, 12-Feb-2008.)
Assertion
Ref Expression
fveleq (𝐴 = 𝐵 → ((𝜑 → (𝐹𝐴) ∈ 𝑃) ↔ (𝜑 → (𝐹𝐵) ∈ 𝑃)))

Proof of Theorem fveleq
StepHypRef Expression
1 fveq2 5986 . . 3 (𝐴 = 𝐵 → (𝐹𝐴) = (𝐹𝐵))
21eleq1d 2576 . 2 (𝐴 = 𝐵 → ((𝐹𝐴) ∈ 𝑃 ↔ (𝐹𝐵) ∈ 𝑃))
32imbi2d 328 1 (𝐴 = 𝐵 → ((𝜑 → (𝐹𝐴) ∈ 𝑃) ↔ (𝜑 → (𝐹𝐵) ∈ 𝑃)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194   = wceq 1474  wcel 1938  cfv 5689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-rex 2806  df-rab 2809  df-v 3079  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-nul 3778  df-if 3940  df-sn 4029  df-pr 4031  df-op 4035  df-uni 4271  df-br 4482  df-iota 5653  df-fv 5697
This theorem is referenced by:  findfvcl  31485
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