Theorem mpteq1OLD 5181

Description: Obsolete version of mpteq1 5180 as of 11-Nov-2024. (Contributed by Mario Carneiro, 16-Dec-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
mpteq1OLD	⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem mpteq1OLD

Step	Hyp	Ref	Expression
1		eqidd 2738	. . 3 ⊢ (𝑥 ∈ 𝐴 → 𝐶 = 𝐶)
2	1	rgen 3064	. 2 ⊢ ∀𝑥 ∈ 𝐴 𝐶 = 𝐶
3		mpteq12 5179	. 2 ⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝐶 = 𝐶) → (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶))
4	2, 3	mpan2 688	1 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2105  ∀wral 3062   ↦ cmpt 5170

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-12 2170  ax-ext 2708

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1543  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3063  df-opab 5150  df-mpt 5171

This theorem is referenced by: (None)