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Mirrors > Home > MPE Home > Th. List > Mathboxes > mpteq12daOLD | Structured version Visualization version GIF version |
Description: Obsolete version of mpteq12da 5232 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 2210 | . 2 ⊢ (𝜑 → ∀𝑥 𝐴 = 𝐶) |
4 | mpteq12daOLD.3 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐷) | |
5 | 1, 4 | ralrimia 3255 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 = 𝐷) |
6 | mpteq12f 5235 | . 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 1534 = wceq 1536 Ⅎwnf 1779 ∈ wcel 2105 ∀wral 3058 ↦ cmpt 5230 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-12 2174 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1539 df-ex 1776 df-nf 1780 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ral 3059 df-opab 5210 df-mpt 5231 |
This theorem is referenced by: (None) |
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