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Mirrors > Home > MPE Home > Th. List > csbco3g | Structured version Visualization version GIF version |
Description: Composition of two class substitutions. Usage of this theorem is discouraged because it depends on ax-13 2371. (Contributed by NM, 27-Nov-2005.) (Revised by Mario Carneiro, 11-Nov-2016.) (New usage is discouraged.) |
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 4387 | . 2 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐷 = ⦋⦋𝐴 / 𝑥⦌𝐵 / 𝑦⦌𝐷) | |
2 | elex 3462 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
3 | nfcvd 2905 | . . . . 5 ⊢ (𝐴 ∈ V → Ⅎ𝑥𝐶) | |
4 | sbcco3g.1 | . . . . 5 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) | |
5 | 3, 4 | csbiegf 3890 | . . . 4 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = 𝐶) |
6 | 2, 5 | syl 17 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐶) |
7 | 6 | csbeq1d 3860 | . 2 ⊢ (𝐴 ∈ 𝑉 → ⦋⦋𝐴 / 𝑥⦌𝐵 / 𝑦⦌𝐷 = ⦋𝐶 / 𝑦⦌𝐷) |
8 | 1, 7 | eqtrd 2773 | 1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐷 = ⦋𝐶 / 𝑦⦌𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 Vcvv 3444 ⦋csb 3856 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-13 2371 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-v 3446 df-sbc 3741 df-csb 3857 |
This theorem is referenced by: (None) |
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