Mathbox for Giovanni Mascellani |
< Previous
Next >
Nearby theorems |
||
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 33918 | . 2 ⊢ ([𝐴 / 𝑥]∃𝑦𝜑 ↔ ∃𝑦[𝐴 / 𝑥]𝜑) |
3 | sbcexfi.2 | . . 3 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜓) | |
4 | 3 | exbii 1774 | . 2 ⊢ (∃𝑦[𝐴 / 𝑥]𝜑 ↔ ∃𝑦𝜓) |
5 | 2, 4 | bitri 264 | 1 ⊢ ([𝐴 / 𝑥]∃𝑦𝜑 ↔ ∃𝑦𝜓) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∃wex 1704 Ⅎwnfc 2751 [wsbc 3435 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-v 3202 df-sbc 3436 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |