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| Mirrors > Home > MPE Home > Th. List > Mathboxes > sbcexfi | Structured version Visualization version GIF version | ||
| Description: Move existential quantifier in and out of class substitution, with an explicit nonfree variable condition and in inference form. (Contributed by Giovanni Mascellani, 30-May-2019.) | 
| Ref | Expression | 
|---|---|
| sbcexfi.1 | ⊢ Ⅎ𝑦𝐴 | 
| sbcexfi.2 | ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜓) | 
| Ref | Expression | 
|---|---|
| sbcexfi | ⊢ ([𝐴 / 𝑥]∃𝑦𝜑 ↔ ∃𝑦𝜓) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | sbcexfi.1 | . . 3 ⊢ Ⅎ𝑦𝐴 | |
| 2 | 1 | sbcexf 38122 | . 2 ⊢ ([𝐴 / 𝑥]∃𝑦𝜑 ↔ ∃𝑦[𝐴 / 𝑥]𝜑) | 
| 3 | sbcexfi.2 | . . 3 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜓) | |
| 4 | 3 | exbii 1848 | . 2 ⊢ (∃𝑦[𝐴 / 𝑥]𝜑 ↔ ∃𝑦𝜓) | 
| 5 | 2, 4 | bitri 275 | 1 ⊢ ([𝐴 / 𝑥]∃𝑦𝜑 ↔ ∃𝑦𝜓) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∃wex 1779 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: (None) | 
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