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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 3912 | . 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 208 ∀wal 1535 = wceq 1537 ∈ wcel 2114 Ⅎwnfc 2961 Vcvv 3494 ⦋csb 3883 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3496 df-sbc 3773 df-csb 3884 |
This theorem is referenced by: csbiebg 3915 |
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