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Mirrors > Home > MPE Home > Th. List > sbcimdvOLD | Structured version Visualization version GIF version |
Description: Obsolete proof of sbcimdv 3498 as of 7-Jul-2021. (Contributed by NM, 11-Nov-2005.) (Revised by NM, 17-Aug-2018.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
sbcimdv.1 | ⊢ (𝜑 → (𝜓 → 𝜒)) |
Ref | Expression |
---|---|
sbcimdvOLD | ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbcimdv.1 | . . . 4 ⊢ (𝜑 → (𝜓 → 𝜒)) | |
2 | 1 | alrimiv 1855 | . . 3 ⊢ (𝜑 → ∀𝑥(𝜓 → 𝜒)) |
3 | spsbc 3448 | . . 3 ⊢ (𝐴 ∈ V → (∀𝑥(𝜓 → 𝜒) → [𝐴 / 𝑥](𝜓 → 𝜒))) | |
4 | sbcim1 3482 | . . 3 ⊢ ([𝐴 / 𝑥](𝜓 → 𝜒) → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)) | |
5 | 2, 3, 4 | syl56 36 | . 2 ⊢ (𝐴 ∈ V → (𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒))) |
6 | sbcex 3445 | . . . . 5 ⊢ ([𝐴 / 𝑥]𝜓 → 𝐴 ∈ V) | |
7 | 6 | con3i 150 | . . . 4 ⊢ (¬ 𝐴 ∈ V → ¬ [𝐴 / 𝑥]𝜓) |
8 | 7 | pm2.21d 118 | . . 3 ⊢ (¬ 𝐴 ∈ V → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)) |
9 | 8 | a1d 25 | . 2 ⊢ (¬ 𝐴 ∈ V → (𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒))) |
10 | 5, 9 | pm2.61i 176 | 1 ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1481 ∈ wcel 1990 Vcvv 3200 [wsbc 3435 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-v 3202 df-sbc 3436 |
This theorem is referenced by: (None) |
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