Theorem fveleq 36495

Description: Please add description here. (Contributed by Jeff Hoffman, 12-Feb-2008.)

Assertion

Ref	Expression
fveleq	⊢ (𝐴 = 𝐵 → ((𝜑 → (𝐹‘𝐴) ∈ 𝑃) ↔ (𝜑 → (𝐹‘𝐵) ∈ 𝑃)))

Proof of Theorem fveleq

Step	Hyp	Ref	Expression
1		fveq2 6822	. . 3 ⊢ (𝐴 = 𝐵 → (𝐹‘𝐴) = (𝐹‘𝐵))
2	1	eleq1d 2816	. 2 ⊢ (𝐴 = 𝐵 → ((𝐹‘𝐴) ∈ 𝑃 ↔ (𝐹‘𝐵) ∈ 𝑃))
3	2	imbi2d 340	1 ⊢ (𝐴 = 𝐵 → ((𝜑 → (𝐹‘𝐴) ∈ 𝑃) ↔ (𝜑 → (𝐹‘𝐵) ∈ 𝑃)))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1541   ∈ wcel 2111  ‘cfv 6481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-iota 6437  df-fv 6489

This theorem is referenced by:  findfvcl  36496