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Mirrors > Home > ILE Home > Th. List > sbcimdv | GIF version |
Description: Substitution analogue of Theorem 19.20 of [Margaris] p. 90 (alim 1457). (Contributed by NM, 11-Nov-2005.) (Revised by NM, 17-Aug-2018.) (Proof shortened by JJ, 7-Jul-2021.) |
Ref | Expression |
---|---|
sbcimdv.1 | ⊢ (𝜑 → (𝜓 → 𝜒)) |
Ref | Expression |
---|---|
sbcimdv | ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbcex 2971 | . 2 ⊢ ([𝐴 / 𝑥]𝜓 → 𝐴 ∈ V) | |
2 | sbcimdv.1 | . . . . 5 ⊢ (𝜑 → (𝜓 → 𝜒)) | |
3 | 2 | alrimiv 1874 | . . . 4 ⊢ (𝜑 → ∀𝑥(𝜓 → 𝜒)) |
4 | spsbc 2974 | . . . 4 ⊢ (𝐴 ∈ V → (∀𝑥(𝜓 → 𝜒) → [𝐴 / 𝑥](𝜓 → 𝜒))) | |
5 | sbcim1 3011 | . . . 4 ⊢ ([𝐴 / 𝑥](𝜓 → 𝜒) → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)) | |
6 | 3, 4, 5 | syl56 34 | . . 3 ⊢ (𝐴 ∈ V → (𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒))) |
7 | 6 | com3l 81 | . 2 ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 → (𝐴 ∈ V → [𝐴 / 𝑥]𝜒))) |
8 | 1, 7 | mpdi 43 | 1 ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1351 ∈ wcel 2148 Vcvv 2737 [wsbc 2962 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-v 2739 df-sbc 2963 |
This theorem is referenced by: (None) |
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