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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 7494 | 1 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑤, 𝑣⟩, 𝑧⟩ ∣ 𝜓} |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 {coprab 7406 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-opab 5204 df-oprab 7409 |
This theorem is referenced by: cpnnen 16179 |
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