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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 2353 | . . 3 ⊢ (∀𝑦𝜑 ↔ ∀𝑧[𝑧 / 𝑦]𝜑) | |
2 | 1 | sbcbii 3852 | . 2 ⊢ ([𝐴 / 𝑥]∀𝑦𝜑 ↔ [𝐴 / 𝑥]∀𝑧[𝑧 / 𝑦]𝜑) |
3 | sbcal 3855 | . 2 ⊢ ([𝐴 / 𝑥]∀𝑧[𝑧 / 𝑦]𝜑 ↔ ∀𝑧[𝐴 / 𝑥][𝑧 / 𝑦]𝜑) | |
4 | sbcalf.1 | . . . 4 ⊢ Ⅎ𝑦𝐴 | |
5 | nfs1v 2154 | . . . 4 ⊢ Ⅎ𝑦[𝑧 / 𝑦]𝜑 | |
6 | 4, 5 | nfsbcw 3813 | . . 3 ⊢ Ⅎ𝑦[𝐴 / 𝑥][𝑧 / 𝑦]𝜑 |
7 | nfv 1912 | . . 3 ⊢ Ⅎ𝑧[𝐴 / 𝑥]𝜑 | |
8 | sbequ12r 2250 | . . . 4 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑦]𝜑 ↔ 𝜑)) | |
9 | 8 | sbcbidv 3851 | . . 3 ⊢ (𝑧 = 𝑦 → ([𝐴 / 𝑥][𝑧 / 𝑦]𝜑 ↔ [𝐴 / 𝑥]𝜑)) |
10 | 6, 7, 9 | cbvalv1 2342 | . 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 1535 [wsb 2062 Ⅎwnfc 2888 [wsbc 3791 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-v 3480 df-sbc 3792 |
This theorem is referenced by: sbcalfi 38103 |
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