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| Mirrors > Home > MPE Home > Th. List > mpteq2daOLD | Structured version Visualization version GIF version | ||
| Description: Obsolete version of mpteq2da 5239 as of 11-Nov-2024. (Contributed by FL, 14-Sep-2013.) (Revised by Mario Carneiro, 16-Dec-2013.) (Proof modification is discouraged.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| mpteq2da.1 | ⊢ Ⅎ𝑥𝜑 | 
| mpteq2da.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶) | 
| Ref | Expression | 
|---|---|
| mpteq2daOLD | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eqid 2736 | . . 3 ⊢ 𝐴 = 𝐴 | |
| 2 | 1 | ax-gen 1794 | . 2 ⊢ ∀𝑥 𝐴 = 𝐴 | 
| 3 | mpteq2da.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
| 4 | mpteq2da.2 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶) | |
| 5 | 4 | ex 412 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐵 = 𝐶)) | 
| 6 | 3, 5 | ralrimi 3256 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 = 𝐶) | 
| 7 | mpteq12f 5229 | . 2 ⊢ ((∀𝑥 𝐴 = 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐶) → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶)) | |
| 8 | 2, 6, 7 | sylancr 587 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∀wal 1537 = wceq 1539 Ⅎwnf 1782 ∈ wcel 2107 ∀wral 3060 ↦ cmpt 5224 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-12 2176 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1542 df-ex 1779 df-nf 1783 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-opab 5205 df-mpt 5225 | 
| This theorem is referenced by: (None) | 
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