Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > csbiebg | 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, 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 2633 | . . . 4 ⊢ (𝑎 = 𝐴 → (𝑥 = 𝑎 ↔ 𝑥 = 𝐴)) | |
2 | 1 | imbi1d 331 | . . 3 ⊢ (𝑎 = 𝐴 → ((𝑥 = 𝑎 → 𝐵 = 𝐶) ↔ (𝑥 = 𝐴 → 𝐵 = 𝐶))) |
3 | 2 | albidv 1849 | . 2 ⊢ (𝑎 = 𝐴 → (∀𝑥(𝑥 = 𝑎 → 𝐵 = 𝐶) ↔ ∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶))) |
4 | csbeq1 3536 | . . 3 ⊢ (𝑎 = 𝐴 → ⦋𝑎 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵) | |
5 | 4 | eqeq1d 2624 | . 2 ⊢ (𝑎 = 𝐴 → (⦋𝑎 / 𝑥⦌𝐵 = 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶)) |
6 | vex 3203 | . . 3 ⊢ 𝑎 ∈ V | |
7 | csbiebg.2 | . . 3 ⊢ Ⅎ𝑥𝐶 | |
8 | 6, 7 | csbieb 3555 | . 2 ⊢ (∀𝑥(𝑥 = 𝑎 → 𝐵 = 𝐶) ↔ ⦋𝑎 / 𝑥⦌𝐵 = 𝐶) |
9 | 3, 5, 8 | vtoclbg 3267 | 1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∀wal 1481 = wceq 1483 ∈ wcel 1990 Ⅎwnfc 2751 ⦋csb 3533 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-v 3202 df-sbc 3436 df-csb 3534 |
This theorem is referenced by: cdlemefrs29bpre0 35684 |
Copyright terms: Public domain | W3C validator |