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Mirrors > Home > MPE Home > Th. List > Mathboxes > sbcimi | Structured version Visualization version GIF version |
Description: Distribution of class substitution over implication, in inference form. (Contributed by Giovanni Mascellani, 27-May-2019.) |
Ref | Expression |
---|---|
sbcimi.1 | ⊢ 𝐴 ∈ V |
sbcimi.2 | ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜒) |
sbcimi.3 | ⊢ ([𝐴 / 𝑥]𝜓 ↔ 𝜂) |
Ref | Expression |
---|---|
sbcimi | ⊢ ([𝐴 / 𝑥](𝜑 → 𝜓) ↔ (𝜒 → 𝜂)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbcimi.1 | . . 3 ⊢ 𝐴 ∈ V | |
2 | sbcimg 3771 | . . 3 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥](𝜑 → 𝜓) ↔ ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓))) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ ([𝐴 / 𝑥](𝜑 → 𝜓) ↔ ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓)) |
4 | sbcimi.2 | . . 3 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜒) | |
5 | sbcimi.3 | . . 3 ⊢ ([𝐴 / 𝑥]𝜓 ↔ 𝜂) | |
6 | 4, 5 | imbi12i 351 | . 2 ⊢ (([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓) ↔ (𝜒 → 𝜂)) |
7 | 3, 6 | bitri 274 | 1 ⊢ ([𝐴 / 𝑥](𝜑 → 𝜓) ↔ (𝜒 → 𝜂)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∈ wcel 2110 Vcvv 3431 [wsbc 3720 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-12 2175 ax-ext 2711 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-tru 1545 df-ex 1787 df-nf 1791 df-sb 2072 df-clab 2718 df-cleq 2732 df-clel 2818 df-sbc 3721 |
This theorem is referenced by: (None) |
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