Theorem fveleq

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

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

Proof

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