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Mirrors > Home > MPE Home > Th. List > cbvrexsvw | Structured version Visualization version GIF version |
Description: Change bound variable by using a substitution. Version of cbvrexsv 3339 with a disjoint variable condition, which does not require ax-13 2371. (Contributed by NM, 2-Mar-2008.) Avoid ax-13 2371. (Revised by Gino Giotto, 10-Jan-2024.) |
Ref | Expression |
---|---|
cbvrexsvw | ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 [𝑦 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1918 | . . 3 ⊢ Ⅎ𝑧𝜑 | |
2 | nfs1v 2154 | . . 3 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑 | |
3 | sbequ12 2244 | . . 3 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑)) | |
4 | 1, 2, 3 | cbvrexw 3289 | . 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑧 ∈ 𝐴 [𝑧 / 𝑥]𝜑) |
5 | nfv 1918 | . . 3 ⊢ Ⅎ𝑦[𝑧 / 𝑥]𝜑 | |
6 | nfv 1918 | . . 3 ⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑 | |
7 | sbequ 2087 | . . 3 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)) | |
8 | 5, 6, 7 | cbvrexw 3289 | . 2 ⊢ (∃𝑧 ∈ 𝐴 [𝑧 / 𝑥]𝜑 ↔ ∃𝑦 ∈ 𝐴 [𝑦 / 𝑥]𝜑) |
9 | 4, 8 | bitri 275 | 1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 [𝑦 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 [wsb 2068 ∃wrex 3070 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-10 2138 ax-11 2155 ax-12 2172 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-ex 1783 df-nf 1787 df-sb 2069 df-clel 2811 df-nfc 2886 df-ral 3062 df-rex 3071 |
This theorem is referenced by: rspesbca 3838 ac6sf 10430 ac6gf 36237 |
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