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Mirrors > Home > MPE Home > Th. List > Mathboxes > sbcori | Structured version Visualization version GIF version |
Description: Distribution of class substitution over disjunction, in inference form. (Contributed by Giovanni Mascellani, 27-May-2019.) |
Ref | Expression |
---|---|
sbcori.1 | ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜒) |
sbcori.2 | ⊢ ([𝐴 / 𝑥]𝜓 ↔ 𝜂) |
Ref | Expression |
---|---|
sbcori | ⊢ ([𝐴 / 𝑥](𝜑 ∨ 𝜓) ↔ (𝜒 ∨ 𝜂)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbcor 3768 | . 2 ⊢ ([𝐴 / 𝑥](𝜑 ∨ 𝜓) ↔ ([𝐴 / 𝑥]𝜑 ∨ [𝐴 / 𝑥]𝜓)) | |
2 | sbcori.1 | . . 3 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜒) | |
3 | sbcori.2 | . . 3 ⊢ ([𝐴 / 𝑥]𝜓 ↔ 𝜂) | |
4 | 2, 3 | orbi12i 912 | . 2 ⊢ (([𝐴 / 𝑥]𝜑 ∨ [𝐴 / 𝑥]𝜓) ↔ (𝜒 ∨ 𝜂)) |
5 | 1, 4 | bitri 274 | 1 ⊢ ([𝐴 / 𝑥](𝜑 ∨ 𝜓) ↔ (𝜒 ∨ 𝜂)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∨ wo 844 [wsbc 3715 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-12 2171 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-tru 1542 df-ex 1783 df-nf 1787 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-v 3431 df-sbc 3716 |
This theorem is referenced by: (None) |
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