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Mirrors > Home > MPE Home > Th. List > Mathboxes > sbcalf | Structured version Visualization version GIF version |
Description: Move universal quantifier in and out of class substitution, with an explicit non-free variable condition. (Contributed by Giovanni Mascellani, 29-May-2019.) |
Ref | Expression |
---|---|
sbcalf.1 | ⊢ Ⅎ𝑦𝐴 |
Ref | Expression |
---|---|
sbcalf | ⊢ ([𝐴 / 𝑥]∀𝑦𝜑 ↔ ∀𝑦[𝐴 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1906 | . . . 4 ⊢ Ⅎ𝑧𝜑 | |
2 | 1 | sb8v 2364 | . . 3 ⊢ (∀𝑦𝜑 ↔ ∀𝑧[𝑧 / 𝑦]𝜑) |
3 | 2 | sbcbii 3826 | . 2 ⊢ ([𝐴 / 𝑥]∀𝑦𝜑 ↔ [𝐴 / 𝑥]∀𝑧[𝑧 / 𝑦]𝜑) |
4 | sbcal 3830 | . 2 ⊢ ([𝐴 / 𝑥]∀𝑧[𝑧 / 𝑦]𝜑 ↔ ∀𝑧[𝐴 / 𝑥][𝑧 / 𝑦]𝜑) | |
5 | sbcalf.1 | . . . 4 ⊢ Ⅎ𝑦𝐴 | |
6 | nfs1v 2264 | . . . 4 ⊢ Ⅎ𝑦[𝑧 / 𝑦]𝜑 | |
7 | 5, 6 | nfsbcw 3791 | . . 3 ⊢ Ⅎ𝑦[𝐴 / 𝑥][𝑧 / 𝑦]𝜑 |
8 | nfv 1906 | . . 3 ⊢ Ⅎ𝑧[𝐴 / 𝑥]𝜑 | |
9 | sbequ12r 2244 | . . . 4 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑦]𝜑 ↔ 𝜑)) | |
10 | 9 | sbcbidv 3824 | . . 3 ⊢ (𝑧 = 𝑦 → ([𝐴 / 𝑥][𝑧 / 𝑦]𝜑 ↔ [𝐴 / 𝑥]𝜑)) |
11 | 7, 8, 10 | cbvalv1 2352 | . 2 ⊢ (∀𝑧[𝐴 / 𝑥][𝑧 / 𝑦]𝜑 ↔ ∀𝑦[𝐴 / 𝑥]𝜑) |
12 | 3, 4, 11 | 3bitri 298 | 1 ⊢ ([𝐴 / 𝑥]∀𝑦𝜑 ↔ ∀𝑦[𝐴 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∀wal 1526 [wsb 2060 Ⅎwnfc 2958 [wsbc 3769 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-v 3494 df-sbc 3770 |
This theorem is referenced by: sbcalfi 35275 |
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