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Mirrors > Home > MPE Home > Th. List > csbieb | Structured version Visualization version GIF version |
Description: Bidirectional conversion between an implicit class substitution hypothesis 𝑥 = 𝐴 → 𝐵 = 𝐶 and its explicit substitution equivalent. (Contributed by NM, 2-Mar-2008.) |
Ref | Expression |
---|---|
csbieb.1 | ⊢ 𝐴 ∈ V |
csbieb.2 | ⊢ Ⅎ𝑥𝐶 |
Ref | Expression |
---|---|
csbieb | ⊢ (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | csbieb.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | csbieb.2 | . 2 ⊢ Ⅎ𝑥𝐶 | |
3 | csbiebt 3867 | . 2 ⊢ ((𝐴 ∈ V ∧ Ⅎ𝑥𝐶) → (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶)) | |
4 | 1, 2, 3 | mp2an 690 | 1 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1537 = wceq 1539 ∈ wcel 2104 Ⅎwnfc 2885 Vcvv 3437 ⦋csb 3837 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3an 1089 df-tru 1542 df-ex 1780 df-nf 1784 df-sb 2066 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-v 3439 df-sbc 3722 df-csb 3838 |
This theorem is referenced by: csbiebg 3870 |
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