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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 non-free 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 35274 | . 2 ⊢ ([𝐴 / 𝑥]∃𝑦𝜑 ↔ ∃𝑦[𝐴 / 𝑥]𝜑) |
3 | sbcexfi.2 | . . 3 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜓) | |
4 | 3 | exbii 1839 | . 2 ⊢ (∃𝑦[𝐴 / 𝑥]𝜑 ↔ ∃𝑦𝜓) |
5 | 2, 4 | bitri 276 | 1 ⊢ ([𝐴 / 𝑥]∃𝑦𝜑 ↔ ∃𝑦𝜓) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∃wex 1771 Ⅎwnfc 2958 [wsbc 3769 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-v 3494 df-sbc 3770 |
This theorem is referenced by: (None) |
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