Theorem mpteq2daOLD

Description: Obsolete version of mpteq2da as of 11-Nov-2024. (Contributed by FL, 14-Sep-2013) (Revised by Mario Carneiro, 16-Dec-2013) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypotheses	mpteq2da.1	⊢ Ⅎ 𝑥 𝜑
	mpteq2da.2	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 = 𝐶 )
Assertion	mpteq2daOLD	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		mpteq2da.1	⊢ Ⅎ 𝑥 𝜑
2		mpteq2da.2	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 = 𝐶 )
3		eqid	⊢ 𝐴 = 𝐴
4	3	ax-gen	⊢ ∀ 𝑥 𝐴 = 𝐴
5	2	ex	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 → 𝐵 = 𝐶 ) )
6	1 5	ralrimi	⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 𝐵 = 𝐶 )
7		mpteq12f	⊢ ( ( ∀ 𝑥 𝐴 = 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝐵 = 𝐶 ) → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) )
8	4 6 7	sylancr	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) )