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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 480 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 ∈ 𝑊) |
4 | sbc2iedf.7 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒)) | |
5 | 4 | anassrs 467 | . . 3 ⊢ (((𝜑 ∧ 𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → (𝜓 ↔ 𝜒)) |
6 | sbc2iedf.2 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
7 | nfv 1912 | . . . 4 ⊢ Ⅎ𝑦 𝑥 = 𝐴 | |
8 | 6, 7 | nfan 1897 | . . 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 3836 | . 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 3836 | 1 ⊢ (𝜑 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜓 ↔ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = 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-or 848 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: rspc2daf 32495 |
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