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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 1843 | . . . 4 ⊢ Ⅎ𝑧𝜑 | |
2 | 1 | sb8 2424 | . . 3 ⊢ (∀𝑦𝜑 ↔ ∀𝑧[𝑧 / 𝑦]𝜑) |
3 | 2 | sbcbii 3491 | . 2 ⊢ ([𝐴 / 𝑥]∀𝑦𝜑 ↔ [𝐴 / 𝑥]∀𝑧[𝑧 / 𝑦]𝜑) |
4 | sbcal 3485 | . 2 ⊢ ([𝐴 / 𝑥]∀𝑧[𝑧 / 𝑦]𝜑 ↔ ∀𝑧[𝐴 / 𝑥][𝑧 / 𝑦]𝜑) | |
5 | sbcalf.1 | . . . 4 ⊢ Ⅎ𝑦𝐴 | |
6 | nfs1v 2437 | . . . 4 ⊢ Ⅎ𝑦[𝑧 / 𝑦]𝜑 | |
7 | 5, 6 | nfsbc 3457 | . . 3 ⊢ Ⅎ𝑦[𝐴 / 𝑥][𝑧 / 𝑦]𝜑 |
8 | nfv 1843 | . . 3 ⊢ Ⅎ𝑧[𝐴 / 𝑥]𝜑 | |
9 | sbequ12r 2112 | . . . 4 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑦]𝜑 ↔ 𝜑)) | |
10 | 9 | sbcbidv 3490 | . . 3 ⊢ (𝑧 = 𝑦 → ([𝐴 / 𝑥][𝑧 / 𝑦]𝜑 ↔ [𝐴 / 𝑥]𝜑)) |
11 | 7, 8, 10 | cbval 2271 | . 2 ⊢ (∀𝑧[𝐴 / 𝑥][𝑧 / 𝑦]𝜑 ↔ ∀𝑦[𝐴 / 𝑥]𝜑) |
12 | 3, 4, 11 | 3bitri 286 | 1 ⊢ ([𝐴 / 𝑥]∀𝑦𝜑 ↔ ∀𝑦[𝐴 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∀wal 1481 [wsb 1880 Ⅎwnfc 2751 [wsbc 3435 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-v 3202 df-sbc 3436 |
This theorem is referenced by: sbcalfi 33919 |
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