Theorem fveleq 34567

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 6756	. . 3 ⊢ (𝐴 = 𝐵 → (𝐹‘𝐴) = (𝐹‘𝐵))
2	1	eleq1d 2823	. 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 205   = wceq 1539   ∈ wcel 2108  ‘cfv 6418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-iota 6376  df-fv 6426

This theorem is referenced by:  findfvcl  34568