Theorem mpteq12dvaOLD 5235

Description: Obsolete version of mpteq12dva 5234 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 1923	. 2 ⊢ (𝜑 → ∀𝑥 𝐴 = 𝐶)
3		mpteq12dva.2	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐷)
4	3	ralrimiva 3136	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 = 𝐷)
5		mpteq12f 5233	. 2 ⊢ ((∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐷) → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷))
6	2, 4, 5	syl2anc 582	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷))

Syntax hints:   → wi 4   ∧ wa 394  ∀wal 1532   = wceq 1534   ∈ wcel 2099  ∀wral 3051   ↦ cmpt 5228

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-12 2167  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1537  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ral 3052  df-opab 5208  df-mpt 5229

This theorem is referenced by: (None)