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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 412 | . . 3 ⊢ (𝜑 → (𝑥 = 𝐴 → (𝜓 ↔ 𝜒))) |
6 | 3, 5 | alrimi 2211 | . 2 ⊢ (𝜑 → ∀𝑥(𝑥 = 𝐴 → (𝜓 ↔ 𝜒))) |
7 | sbciegft 3829 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝜒 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜓 ↔ 𝜒))) → ([𝐴 / 𝑥]𝜓 ↔ 𝜒)) | |
8 | 1, 2, 6, 7 | syl3anc 1370 | 1 ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∀wal 1535 = wceq 1537 Ⅎwnf 1780 ∈ wcel 2106 [wsbc 3791 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1088 df-tru 1540 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-sbc 3792 |
This theorem is referenced by: sbciedOLD 3838 sbc2iegf 3873 csbiebt 3938 sbcnestgfw 4427 sbcnestgf 4432 ovmpodxf 7583 sbc2iedf 32494 reuf1odnf 47057 ovmpordxf 48184 |
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