Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > sbc2iedf | Structured version Visualization version GIF version |
Description: Conversion of implicit substitution to explicit class substitution. (Contributed by Thierry Arnoux, 4-Jul-2023.) |
Ref | Expression |
---|---|
sbc2iedf.1 | ⊢ Ⅎ𝑥𝜑 |
sbc2iedf.2 | ⊢ Ⅎ𝑦𝜑 |
sbc2iedf.3 | ⊢ Ⅎ𝑥𝜒 |
sbc2iedf.4 | ⊢ Ⅎ𝑦𝜒 |
sbc2iedf.5 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
sbc2iedf.6 | ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
sbc2iedf.7 | ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
sbc2iedf | ⊢ (𝜑 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜓 ↔ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbc2iedf.5 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
2 | sbc2iedf.6 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
3 | 2 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 ∈ 𝑊) |
4 | sbc2iedf.7 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒)) | |
5 | 4 | anassrs 470 | . . 3 ⊢ (((𝜑 ∧ 𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → (𝜓 ↔ 𝜒)) |
6 | sbc2iedf.2 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
7 | nfv 1914 | . . . 4 ⊢ Ⅎ𝑦 𝑥 = 𝐴 | |
8 | 6, 7 | nfan 1899 | . . 3 ⊢ Ⅎ𝑦(𝜑 ∧ 𝑥 = 𝐴) |
9 | sbc2iedf.4 | . . . 4 ⊢ Ⅎ𝑦𝜒 | |
10 | 9 | a1i 11 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → Ⅎ𝑦𝜒) |
11 | 3, 5, 8, 10 | sbciedf 3809 | . 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → ([𝐵 / 𝑦]𝜓 ↔ 𝜒)) |
12 | sbc2iedf.1 | . 2 ⊢ Ⅎ𝑥𝜑 | |
13 | sbc2iedf.3 | . . 3 ⊢ Ⅎ𝑥𝜒 | |
14 | 13 | a1i 11 | . 2 ⊢ (𝜑 → Ⅎ𝑥𝜒) |
15 | 1, 11, 12, 14 | sbciedf 3809 | 1 ⊢ (𝜑 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜓 ↔ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1536 Ⅎwnf 1783 ∈ wcel 2113 [wsbc 3768 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-12 2176 ax-ext 2792 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-v 3493 df-sbc 3769 |
This theorem is referenced by: rspc2daf 30229 |
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