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Mirrors > Home > MPE Home > Th. List > mpteq2daOLD | Structured version Visualization version GIF version |
Description: Obsolete version of mpteq2da 5246 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 2735 | . . 3 ⊢ 𝐴 = 𝐴 | |
2 | 1 | ax-gen 1792 | . 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 3255 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 = 𝐶) |
7 | mpteq12f 5236 | . 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 1535 = wceq 1537 Ⅎwnf 1780 ∈ wcel 2106 ∀wral 3059 ↦ cmpt 5231 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-opab 5211 df-mpt 5232 |
This theorem is referenced by: (None) |
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