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Mirrors > Home > MPE Home > Th. List > cbvralsvwOLD | Structured version Visualization version GIF version |
Description: Obsolete version of cbvralsvw 3305 as of 8-Mar-2025. (Contributed by NM, 20-Nov-2005.) Avoid ax-13 2365. (Revised by Gino Giotto, 10-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
cbvralsvwOLD | ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 [𝑦 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1909 | . . 3 ⊢ Ⅎ𝑧𝜑 | |
2 | nfs1v 2145 | . . 3 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑 | |
3 | sbequ12 2238 | . . 3 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑)) | |
4 | 1, 2, 3 | cbvralw 3294 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑧 ∈ 𝐴 [𝑧 / 𝑥]𝜑) |
5 | nfv 1909 | . . 3 ⊢ Ⅎ𝑦[𝑧 / 𝑥]𝜑 | |
6 | nfv 1909 | . . 3 ⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑 | |
7 | sbequ 2078 | . . 3 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)) | |
8 | 5, 6, 7 | cbvralw 3294 | . 2 ⊢ (∀𝑧 ∈ 𝐴 [𝑧 / 𝑥]𝜑 ↔ ∀𝑦 ∈ 𝐴 [𝑦 / 𝑥]𝜑) |
9 | 4, 8 | bitri 274 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 [𝑦 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 [wsb 2059 ∀wral 3051 |
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-10 2129 ax-11 2146 ax-12 2166 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-ex 1774 df-nf 1778 df-sb 2060 df-clel 2802 df-nfc 2877 df-ral 3052 |
This theorem is referenced by: (None) |
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