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Mirrors > Home > MPE Home > Th. List > cbvopab1vOLD | Structured version Visualization version GIF version |
Description: Obsolete version of cbvopab1v 5221 as of 17-Nov-2024. (Contributed by NM, 31-Jul-2003.) (Proof shortened by Eric Schmidt, 4-Apr-2007.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
cbvopab1vOLD.1 | ⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbvopab1vOLD | ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑧, 𝑦⟩ ∣ 𝜓} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1910 | . 2 ⊢ Ⅎ𝑧𝜑 | |
2 | nfv 1910 | . 2 ⊢ Ⅎ𝑥𝜓 | |
3 | cbvopab1vOLD.1 | . 2 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜓)) | |
4 | 1, 2, 3 | cbvopab1 5217 | 1 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑧, 𝑦⟩ ∣ 𝜓} |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1534 {copab 5204 |
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-ext 2698 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2705 df-cleq 2719 df-clel 2805 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-opab 5205 |
This theorem is referenced by: (None) |
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