Theorem fveleq 33912

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 6645	. . 3 ⊢ (𝐴 = 𝐵 → (𝐹‘𝐴) = (𝐹‘𝐵))
2	1	eleq1d 2874	. 2 ⊢ (𝐴 = 𝐵 → ((𝐹‘𝐴) ∈ 𝑃 ↔ (𝐹‘𝐵) ∈ 𝑃))
3	2	imbi2d 344	1 ⊢ (𝐴 = 𝐵 → ((𝜑 → (𝐹‘𝐴) ∈ 𝑃) ↔ (𝜑 → (𝐹‘𝐵) ∈ 𝑃)))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1538   ∈ wcel 2111  ‘cfv 6324

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-un 3886  df-in 3888  df-ss 3898  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-iota 6283  df-fv 6332

This theorem is referenced by:  findfvcl  33913