Theorem mpteq12dvaOLD 5179

Description: Obsolete version of mpteq12dva 5178 as of 11-Nov-2024. (Contributed by Mario Carneiro, 26-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
mpteq12dv.1	⊢ (𝜑 → 𝐴 = 𝐶)
mpteq12dva.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐷)

Assertion

Ref	Expression
mpteq12dvaOLD	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷))

Distinct variable group:   𝜑,𝑥

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐶(𝑥)   𝐷(𝑥)

Proof of Theorem mpteq12dvaOLD

Step	Hyp	Ref	Expression
1		mpteq12dv.1	. . 3 ⊢ (𝜑 → 𝐴 = 𝐶)
2	1	alrimiv 1929	. 2 ⊢ (𝜑 → ∀𝑥 𝐴 = 𝐶)
3		mpteq12dva.2	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐷)
4	3	ralrimiva 3139	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 = 𝐷)
5		mpteq12f 5177	. 2 ⊢ ((∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐷) → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷))
6	2, 4, 5	syl2anc 584	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷))

Syntax hints:   → wi 4   ∧ wa 396  ∀wal 1538   = wceq 1540   ∈ wcel 2105  ∀wral 3061   ↦ cmpt 5172

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 2707

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 2714  df-cleq 2728  df-clel 2814  df-ral 3062  df-opab 5152  df-mpt 5173

This theorem is referenced by: (None)