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Mirrors > Home > MPE Home > Th. List > sbciedf | Structured version Visualization version GIF version |
Description: Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 29-Dec-2014.) |
Ref | Expression |
---|---|
sbcied.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
sbcied.2 | ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) |
sbciedf.3 | ⊢ Ⅎ𝑥𝜑 |
sbciedf.4 | ⊢ (𝜑 → Ⅎ𝑥𝜒) |
Ref | Expression |
---|---|
sbciedf | ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbcied.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
2 | sbciedf.4 | . 2 ⊢ (𝜑 → Ⅎ𝑥𝜒) | |
3 | sbciedf.3 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
4 | sbcied.2 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) | |
5 | 4 | ex 397 | . . 3 ⊢ (𝜑 → (𝑥 = 𝐴 → (𝜓 ↔ 𝜒))) |
6 | 3, 5 | alrimi 2238 | . 2 ⊢ (𝜑 → ∀𝑥(𝑥 = 𝐴 → (𝜓 ↔ 𝜒))) |
7 | sbciegft 3618 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝜒 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜓 ↔ 𝜒))) → ([𝐴 / 𝑥]𝜓 ↔ 𝜒)) | |
8 | 1, 2, 6, 7 | syl3anc 1476 | 1 ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 382 ∀wal 1629 = wceq 1631 Ⅎwnf 1856 ∈ wcel 2145 [wsbc 3587 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-9 2154 ax-10 2174 ax-12 2203 ax-ext 2751 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-clab 2758 df-cleq 2764 df-clel 2767 df-v 3353 df-sbc 3588 |
This theorem is referenced by: sbcied 3624 sbc2iegf 3654 csbiebt 3702 sbcnestgf 4140 ovmpt2dxf 6937 ovmpt2rdxf 42640 |
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