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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 2213 | . 2 ⊢ (𝜑 → ∀𝑥 𝐴 = 𝐶) |
| 4 | mpteq12daOLD.3 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐷) | |
| 5 | 1, 4 | ralrimia 3258 | . 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 584 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∀wal 1538 = wceq 1540 Ⅎwnf 1783 ∈ wcel 2108 ∀wral 3061 ↦ cmpt 5225 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-opab 5206 df-mpt 5226 |
| This theorem is referenced by: (None) |
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