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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 1444). (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 2954 | . 2 ⊢ ([𝐴 / 𝑥]𝜓 → 𝐴 ∈ V) | |
2 | sbcimdv.1 | . . . . 5 ⊢ (𝜑 → (𝜓 → 𝜒)) | |
3 | 2 | alrimiv 1861 | . . . 4 ⊢ (𝜑 → ∀𝑥(𝜓 → 𝜒)) |
4 | spsbc 2957 | . . . 4 ⊢ (𝐴 ∈ V → (∀𝑥(𝜓 → 𝜒) → [𝐴 / 𝑥](𝜓 → 𝜒))) | |
5 | sbcim1 2994 | . . . 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 1340 ∈ wcel 2135 Vcvv 2721 [wsbc 2946 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-v 2723 df-sbc 2947 |
This theorem is referenced by: (None) |
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