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Mirrors > Home > MPE Home > Th. List > cbvoprab12v | Structured version Visualization version GIF version |
Description: Rule used to change first two bound variables in an operation abstraction, using implicit substitution. (Contributed by NM, 8-Oct-2004.) |
Ref | Expression |
---|---|
cbvoprab12v.1 | ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑣) → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbvoprab12v | ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑤, 𝑣⟩, 𝑧⟩ ∣ 𝜓} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1909 | . 2 ⊢ Ⅎ𝑤𝜑 | |
2 | nfv 1909 | . 2 ⊢ Ⅎ𝑣𝜑 | |
3 | nfv 1909 | . 2 ⊢ Ⅎ𝑥𝜓 | |
4 | nfv 1909 | . 2 ⊢ Ⅎ𝑦𝜓 | |
5 | cbvoprab12v.1 | . 2 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑣) → (𝜑 ↔ 𝜓)) | |
6 | 1, 2, 3, 4, 5 | cbvoprab12 7505 | 1 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑤, 𝑣⟩, 𝑧⟩ ∣ 𝜓} |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 {coprab 7416 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-rab 3420 df-v 3465 df-dif 3943 df-un 3945 df-ss 3957 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-opab 5206 df-oprab 7419 |
This theorem is referenced by: cpnnen 16203 |
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