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Mirrors > Home > ILE Home > Th. List > csbco3g | GIF version |
Description: Composition of two class substitutions. (Contributed by NM, 27-Nov-2005.) (Revised by Mario Carneiro, 11-Nov-2016.) |
Ref | Expression |
---|---|
sbcco3g.1 | ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
csbco3g | ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐷 = ⦋𝐶 / 𝑦⦌𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | csbnestg 2956 | . 2 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐷 = ⦋⦋𝐴 / 𝑥⦌𝐵 / 𝑦⦌𝐷) | |
2 | elex 2610 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
3 | nfcvd 2220 | . . . . 5 ⊢ (𝐴 ∈ V → Ⅎ𝑥𝐶) | |
4 | sbcco3g.1 | . . . . 5 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) | |
5 | 3, 4 | csbiegf 2946 | . . . 4 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = 𝐶) |
6 | 2, 5 | syl 14 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐶) |
7 | 6 | csbeq1d 2914 | . 2 ⊢ (𝐴 ∈ 𝑉 → ⦋⦋𝐴 / 𝑥⦌𝐵 / 𝑦⦌𝐷 = ⦋𝐶 / 𝑦⦌𝐷) |
8 | 1, 7 | eqtrd 2113 | 1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐷 = ⦋𝐶 / 𝑦⦌𝐷) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1284 ∈ wcel 1433 Vcvv 2601 ⦋csb 2908 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-v 2603 df-sbc 2816 df-csb 2909 |
This theorem is referenced by: (None) |
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