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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 38101 | . 2 ⊢ ([𝐴 / 𝑥]∃𝑦𝜑 ↔ ∃𝑦[𝐴 / 𝑥]𝜑) |
3 | sbcexfi.2 | . . 3 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜓) | |
4 | 3 | exbii 1844 | . 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 1775 Ⅎwnfc 2887 [wsbc 3790 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1539 df-ex 1776 df-nf 1780 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-v 3479 df-sbc 3791 |
This theorem is referenced by: (None) |
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