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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cbvrabv2 | Structured version Visualization version GIF version |
Description: A more general version of cbvrabv 3437. Usage of this theorem is discouraged because it depends on ax-13 2366. Use of cbvrabv2w 44466 is preferred. (Contributed by Glauco Siliprandi, 23-Oct-2021.) (New usage is discouraged.) |
Ref | Expression |
---|---|
cbvrabv2.1 | ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) |
cbvrabv2.2 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbvrabv2 | ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∈ 𝐵 ∣ 𝜓} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2898 | . 2 ⊢ Ⅎ𝑦𝐴 | |
2 | nfcv 2898 | . 2 ⊢ Ⅎ𝑥𝐵 | |
3 | nfv 1910 | . 2 ⊢ Ⅎ𝑦𝜑 | |
4 | nfv 1910 | . 2 ⊢ Ⅎ𝑥𝜓 | |
5 | cbvrabv2.1 | . 2 ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) | |
6 | cbvrabv2.2 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
7 | 1, 2, 3, 4, 5, 6 | cbvrabcsf 3937 | 1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∈ 𝐵 ∣ 𝜓} |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1534 {crab 3427 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-13 2366 ax-ext 2698 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-tru 1537 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-rab 3428 df-sbc 3775 df-csb 3890 |
This theorem is referenced by: (None) |
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