![]() |
Mathbox for Giovanni Mascellani |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > sbcexf | Structured version Visualization version GIF version |
Description: Move existential quantifier in and out of class substitution, with an explicit nonfree variable condition. (Contributed by Giovanni Mascellani, 29-May-2019.) |
Ref | Expression |
---|---|
sbcexf.1 | ⊢ Ⅎ𝑦𝐴 |
Ref | Expression |
---|---|
sbcexf | ⊢ ([𝐴 / 𝑥]∃𝑦𝜑 ↔ ∃𝑦[𝐴 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1910 | . . . 4 ⊢ Ⅎ𝑧𝜑 | |
2 | 1 | sb8ef 2346 | . . 3 ⊢ (∃𝑦𝜑 ↔ ∃𝑧[𝑧 / 𝑦]𝜑) |
3 | 2 | sbcbii 3837 | . 2 ⊢ ([𝐴 / 𝑥]∃𝑦𝜑 ↔ [𝐴 / 𝑥]∃𝑧[𝑧 / 𝑦]𝜑) |
4 | sbcex2 3841 | . 2 ⊢ ([𝐴 / 𝑥]∃𝑧[𝑧 / 𝑦]𝜑 ↔ ∃𝑧[𝐴 / 𝑥][𝑧 / 𝑦]𝜑) | |
5 | sbcexf.1 | . . . 4 ⊢ Ⅎ𝑦𝐴 | |
6 | nfs1v 2146 | . . . 4 ⊢ Ⅎ𝑦[𝑧 / 𝑦]𝜑 | |
7 | 5, 6 | nfsbcw 3798 | . . 3 ⊢ Ⅎ𝑦[𝐴 / 𝑥][𝑧 / 𝑦]𝜑 |
8 | nfv 1910 | . . 3 ⊢ Ⅎ𝑧[𝐴 / 𝑥]𝜑 | |
9 | sbequ12r 2240 | . . . 4 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑦]𝜑 ↔ 𝜑)) | |
10 | 9 | sbcbidv 3836 | . . 3 ⊢ (𝑧 = 𝑦 → ([𝐴 / 𝑥][𝑧 / 𝑦]𝜑 ↔ [𝐴 / 𝑥]𝜑)) |
11 | 7, 8, 10 | cbvexv1 2333 | . 2 ⊢ (∃𝑧[𝐴 / 𝑥][𝑧 / 𝑦]𝜑 ↔ ∃𝑦[𝐴 / 𝑥]𝜑) |
12 | 3, 4, 11 | 3bitri 296 | 1 ⊢ ([𝐴 / 𝑥]∃𝑦𝜑 ↔ ∃𝑦[𝐴 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∃wex 1774 [wsb 2060 Ⅎwnfc 2876 [wsbc 3776 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-tru 1537 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-v 3464 df-sbc 3777 |
This theorem is referenced by: sbcexfi 37818 |
Copyright terms: Public domain | W3C validator |