![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 1793 | . 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 3949 | . 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 395 ∀wal 1535 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ⦋csb 3908 |
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-10 2139 ax-11 2155 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-nfc 2890 df-v 3480 df-sbc 3792 df-csb 3909 |
This theorem is referenced by: fsumcnv 15806 fprodcnv 16016 rnghmval 20457 dfrhm2 20491 mamufval 22412 mvmulfval 22564 vtxdgfval 29500 grtri 47845 |
Copyright terms: Public domain | W3C validator |