Mathbox for Alan Sare |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > sbcim2g | Structured version Visualization version GIF version |
Description: Distribution of class substitution over a left-nested implication. Similar to sbcimg 3819. sbcim2g 40865 is sbcim2gVD 41202 without virtual deductions and was automatically derived from sbcim2gVD 41202 using the tools program translate..without..overwriting.cmd and Metamath's minimize command. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
sbcim2g | ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒)) ↔ ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbcimg 3819 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒)) ↔ ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥](𝜓 → 𝜒)))) | |
2 | 1 | biimpd 231 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒)) → ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥](𝜓 → 𝜒)))) |
3 | sbcimg 3819 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝜓 → 𝜒) ↔ ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒))) | |
4 | imbi2 351 | . . . 4 ⊢ (([𝐴 / 𝑥](𝜓 → 𝜒) ↔ ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)) → (([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥](𝜓 → 𝜒)) ↔ ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)))) | |
5 | 4 | biimpcd 251 | . . 3 ⊢ (([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥](𝜓 → 𝜒)) → (([𝐴 / 𝑥](𝜓 → 𝜒) ↔ ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)) → ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)))) |
6 | 2, 3, 5 | syl6ci 71 | . 2 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒)) → ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)))) |
7 | idd 24 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)) → ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)))) | |
8 | biimpr 222 | . . . 4 ⊢ (([𝐴 / 𝑥](𝜓 → 𝜒) ↔ ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)) → (([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒) → [𝐴 / 𝑥](𝜓 → 𝜒))) | |
9 | 3, 7, 8 | ee13 40831 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)) → ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥](𝜓 → 𝜒)))) |
10 | 9, 1 | sylibrd 261 | . 2 ⊢ (𝐴 ∈ 𝑉 → (([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)) → [𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒)))) |
11 | 6, 10 | impbid 214 | 1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒)) ↔ ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∈ wcel 2110 [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-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-sbc 3772 |
This theorem is referenced by: trsbc 40867 trsbcVD 41204 |
Copyright terms: Public domain | W3C validator |