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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 1714 | . 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 3528 | . 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 383 ∀wal 1473 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ⦋csb 3499 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-sbc 3403 df-csb 3500 |
This theorem is referenced by: fsumcnv 14346 fprodcnv 14552 dfrhm2 18540 mamufval 20010 mvmulfval 20167 vtxdgfval 40683 rnghmval 41681 |
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