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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 non-free 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 1915 | . . . 4 ⊢ Ⅎ𝑧𝜑 | |
2 | 1 | sb8ev 2374 | . . 3 ⊢ (∃𝑦𝜑 ↔ ∃𝑧[𝑧 / 𝑦]𝜑) |
3 | 2 | sbcbii 3831 | . 2 ⊢ ([𝐴 / 𝑥]∃𝑦𝜑 ↔ [𝐴 / 𝑥]∃𝑧[𝑧 / 𝑦]𝜑) |
4 | sbcex2 3836 | . 2 ⊢ ([𝐴 / 𝑥]∃𝑧[𝑧 / 𝑦]𝜑 ↔ ∃𝑧[𝐴 / 𝑥][𝑧 / 𝑦]𝜑) | |
5 | sbcexf.1 | . . . 4 ⊢ Ⅎ𝑦𝐴 | |
6 | nfs1v 2160 | . . . 4 ⊢ Ⅎ𝑦[𝑧 / 𝑦]𝜑 | |
7 | 5, 6 | nfsbcw 3796 | . . 3 ⊢ Ⅎ𝑦[𝐴 / 𝑥][𝑧 / 𝑦]𝜑 |
8 | nfv 1915 | . . 3 ⊢ Ⅎ𝑧[𝐴 / 𝑥]𝜑 | |
9 | sbequ12r 2254 | . . . 4 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑦]𝜑 ↔ 𝜑)) | |
10 | 9 | sbcbidv 3829 | . . 3 ⊢ (𝑧 = 𝑦 → ([𝐴 / 𝑥][𝑧 / 𝑦]𝜑 ↔ [𝐴 / 𝑥]𝜑)) |
11 | 7, 8, 10 | cbvexv1 2362 | . 2 ⊢ (∃𝑧[𝐴 / 𝑥][𝑧 / 𝑦]𝜑 ↔ ∃𝑦[𝐴 / 𝑥]𝜑) |
12 | 3, 4, 11 | 3bitri 299 | 1 ⊢ ([𝐴 / 𝑥]∃𝑦𝜑 ↔ ∃𝑦[𝐴 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∃wex 1780 [wsb 2069 Ⅎwnfc 2963 [wsbc 3774 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-v 3498 df-sbc 3775 |
This theorem is referenced by: sbcexfi 35397 |
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