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Mirrors > Home > MPE Home > Th. List > Mathboxes > mpteq12daOLD | Structured version Visualization version GIF version |
Description: Obsolete version of mpteq12da 5227 as of 11-Nov-2024. (Contributed by Glauco Siliprandi, 23-Oct-2021.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
mpteq12daOLD.1 | ⊢ Ⅎ𝑥𝜑 |
mpteq12daOLD.2 | ⊢ (𝜑 → 𝐴 = 𝐶) |
mpteq12daOLD.3 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐷) |
Ref | Expression |
---|---|
mpteq12daOLD | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mpteq12daOLD.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | mpteq12daOLD.2 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐶) | |
3 | 1, 2 | alrimi 2199 | . 2 ⊢ (𝜑 → ∀𝑥 𝐴 = 𝐶) |
4 | mpteq12daOLD.3 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐷) | |
5 | 1, 4 | ralrimia 3250 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 = 𝐷) |
6 | mpteq12f 5230 | . 2 ⊢ ((∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐷) → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷)) | |
7 | 3, 5, 6 | syl2anc 583 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∀wal 1532 = wceq 1534 Ⅎwnf 1778 ∈ wcel 2099 ∀wral 3056 ↦ cmpt 5225 |
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 2164 ax-ext 2698 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-tru 1537 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2705 df-cleq 2719 df-clel 2805 df-ral 3057 df-opab 5205 df-mpt 5226 |
This theorem is referenced by: (None) |
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