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Mirrors > Home > NFE Home > Th. List > csbiebg | GIF version |
Description: Bidirectional conversion between an implicit class substitution hypothesis x = A → B = C and its explicit substitution equivalent. (Contributed by NM, 24-Mar-2013.) (Revised by Mario Carneiro, 11-Dec-2016.) |
Ref | Expression |
---|---|
csbiebg.2 | ⊢ ℲxC |
Ref | Expression |
---|---|
csbiebg | ⊢ (A ∈ V → (∀x(x = A → B = C) ↔ [A / x]B = C)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq2 2362 | . . . 4 ⊢ (a = A → (x = a ↔ x = A)) | |
2 | 1 | imbi1d 308 | . . 3 ⊢ (a = A → ((x = a → B = C) ↔ (x = A → B = C))) |
3 | 2 | albidv 1625 | . 2 ⊢ (a = A → (∀x(x = a → B = C) ↔ ∀x(x = A → B = C))) |
4 | csbeq1 3139 | . . 3 ⊢ (a = A → [a / x]B = [A / x]B) | |
5 | 4 | eqeq1d 2361 | . 2 ⊢ (a = A → ([a / x]B = C ↔ [A / x]B = C)) |
6 | vex 2862 | . . 3 ⊢ a ∈ V | |
7 | csbiebg.2 | . . 3 ⊢ ℲxC | |
8 | 6, 7 | csbieb 3174 | . 2 ⊢ (∀x(x = a → B = C) ↔ [a / x]B = C) |
9 | 3, 5, 8 | vtoclbg 2915 | 1 ⊢ (A ∈ V → (∀x(x = A → B = C) ↔ [A / x]B = C)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∀wal 1540 = wceq 1642 ∈ wcel 1710 Ⅎwnfc 2476 [csb 3136 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3an 936 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-v 2861 df-sbc 3047 df-csb 3137 |
This theorem is referenced by: (None) |
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