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Mirrors > Home > MPE Home > Th. List > sbcimdvOLD | Structured version Visualization version GIF version |
Description: Obsolete version of sbcimdv 3791 as of 12-Oct-2024. (Contributed by NM, 11-Nov-2005.) (Revised by NM, 17-Aug-2018.) (Proof shortened by JJ, 7-Jul-2021.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
sbcimdvOLD.1 | ⊢ (𝜑 → (𝜓 → 𝜒)) |
Ref | Expression |
---|---|
sbcimdvOLD | ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbcex 3727 | . 2 ⊢ ([𝐴 / 𝑥]𝜓 → 𝐴 ∈ V) | |
2 | sbcimdvOLD.1 | . . . . 5 ⊢ (𝜑 → (𝜓 → 𝜒)) | |
3 | 2 | alrimiv 1930 | . . . 4 ⊢ (𝜑 → ∀𝑥(𝜓 → 𝜒)) |
4 | spsbc 3730 | . . . 4 ⊢ (𝐴 ∈ V → (∀𝑥(𝜓 → 𝜒) → [𝐴 / 𝑥](𝜓 → 𝜒))) | |
5 | sbcim1 3773 | . . . 4 ⊢ ([𝐴 / 𝑥](𝜓 → 𝜒) → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)) | |
6 | 3, 4, 5 | syl56 36 | . . 3 ⊢ (𝐴 ∈ V → (𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒))) |
7 | 6 | com3l 89 | . 2 ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 → (𝐴 ∈ V → [𝐴 / 𝑥]𝜒))) |
8 | 1, 7 | mpdi 45 | 1 ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1537 ∈ wcel 2106 Vcvv 3431 [wsbc 3717 |
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-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1542 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-v 3433 df-sbc 3718 |
This theorem is referenced by: (None) |
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