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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 nonfree 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 | sb8v 2355 | . . 3 ⊢ (∀𝑦𝜑 ↔ ∀𝑧[𝑧 / 𝑦]𝜑) | |
| 2 | 1 | sbcbii 3846 | . 2 ⊢ ([𝐴 / 𝑥]∀𝑦𝜑 ↔ [𝐴 / 𝑥]∀𝑧[𝑧 / 𝑦]𝜑) | 
| 3 | sbcal 3849 | . 2 ⊢ ([𝐴 / 𝑥]∀𝑧[𝑧 / 𝑦]𝜑 ↔ ∀𝑧[𝐴 / 𝑥][𝑧 / 𝑦]𝜑) | |
| 4 | sbcalf.1 | . . . 4 ⊢ Ⅎ𝑦𝐴 | |
| 5 | nfs1v 2156 | . . . 4 ⊢ Ⅎ𝑦[𝑧 / 𝑦]𝜑 | |
| 6 | 4, 5 | nfsbcw 3810 | . . 3 ⊢ Ⅎ𝑦[𝐴 / 𝑥][𝑧 / 𝑦]𝜑 | 
| 7 | nfv 1914 | . . 3 ⊢ Ⅎ𝑧[𝐴 / 𝑥]𝜑 | |
| 8 | sbequ12r 2252 | . . . 4 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑦]𝜑 ↔ 𝜑)) | |
| 9 | 8 | sbcbidv 3845 | . . 3 ⊢ (𝑧 = 𝑦 → ([𝐴 / 𝑥][𝑧 / 𝑦]𝜑 ↔ [𝐴 / 𝑥]𝜑)) | 
| 10 | 6, 7, 9 | cbvalv1 2343 | . 2 ⊢ (∀𝑧[𝐴 / 𝑥][𝑧 / 𝑦]𝜑 ↔ ∀𝑦[𝐴 / 𝑥]𝜑) | 
| 11 | 2, 3, 10 | 3bitri 297 | 1 ⊢ ([𝐴 / 𝑥]∀𝑦𝜑 ↔ ∀𝑦[𝐴 / 𝑥]𝜑) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∀wal 1538 [wsb 2064 Ⅎwnfc 2890 [wsbc 3788 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-v 3482 df-sbc 3789 | 
| This theorem is referenced by: sbcalfi 38123 | 
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