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Mirrors > Home > MPE Home > Th. List > Mathboxes > sbcalfi | Structured version Visualization version GIF version |
Description: Move universal 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 |
---|---|
sbcalfi.1 | ⊢ Ⅎ𝑦𝐴 |
sbcalfi.2 | ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜓) |
Ref | Expression |
---|---|
sbcalfi | ⊢ ([𝐴 / 𝑥]∀𝑦𝜑 ↔ ∀𝑦𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbcalfi.1 | . . 3 ⊢ Ⅎ𝑦𝐴 | |
2 | 1 | sbcalf 35391 | . 2 ⊢ ([𝐴 / 𝑥]∀𝑦𝜑 ↔ ∀𝑦[𝐴 / 𝑥]𝜑) |
3 | sbcalfi.2 | . . 3 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜓) | |
4 | 3 | albii 1816 | . 2 ⊢ (∀𝑦[𝐴 / 𝑥]𝜑 ↔ ∀𝑦𝜓) |
5 | 2, 4 | bitri 277 | 1 ⊢ ([𝐴 / 𝑥]∀𝑦𝜑 ↔ ∀𝑦𝜓) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∀wal 1531 Ⅎwnfc 2961 [wsbc 3771 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3496 df-sbc 3772 |
This theorem is referenced by: (None) |
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