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Mirrors > Home > MPE Home > Th. List > csbie2 | Structured version Visualization version GIF version |
Description: Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 27-Aug-2007.) |
Ref | Expression |
---|---|
csbie2t.1 | ⊢ 𝐴 ∈ V |
csbie2t.2 | ⊢ 𝐵 ∈ V |
csbie2.3 | ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷) |
Ref | Expression |
---|---|
csbie2 | ⊢ ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶 = 𝐷 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | csbie2.3 | . . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷) | |
2 | 1 | gen2 1799 | . 2 ⊢ ∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷) |
3 | csbie2t.1 | . . 3 ⊢ 𝐴 ∈ V | |
4 | csbie2t.2 | . . 3 ⊢ 𝐵 ∈ V | |
5 | 3, 4 | csbie2t 3897 | . 2 ⊢ (∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷) → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶 = 𝐷) |
6 | 2, 5 | ax-mp 5 | 1 ⊢ ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶 = 𝐷 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∀wal 1540 = 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-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: fsumcnv 15663 fprodcnv 15871 dfrhm2 20155 mamufval 21750 mvmulfval 21907 vtxdgfval 28457 rnghmval 46275 |
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