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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 2748 | . . . 4 ⊢ (𝑎 = 𝐴 → (𝑥 = 𝑎 ↔ 𝑥 = 𝐴)) | |
2 | 1 | imbi1d 345 | . . 3 ⊢ (𝑎 = 𝐴 → ((𝑥 = 𝑎 → 𝐵 = 𝐶) ↔ (𝑥 = 𝐴 → 𝐵 = 𝐶))) |
3 | 2 | albidv 1928 | . 2 ⊢ (𝑎 = 𝐴 → (∀𝑥(𝑥 = 𝑎 → 𝐵 = 𝐶) ↔ ∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶))) |
4 | csbeq1 3801 | . . 3 ⊢ (𝑎 = 𝐴 → ⦋𝑎 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵) | |
5 | 4 | eqeq1d 2738 | . 2 ⊢ (𝑎 = 𝐴 → (⦋𝑎 / 𝑥⦌𝐵 = 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶)) |
6 | vex 3402 | . . 3 ⊢ 𝑎 ∈ V | |
7 | csbiebg.2 | . . 3 ⊢ Ⅎ𝑥𝐶 | |
8 | 6, 7 | csbieb 3830 | . 2 ⊢ (∀𝑥(𝑥 = 𝑎 → 𝐵 = 𝐶) ↔ ⦋𝑎 / 𝑥⦌𝐵 = 𝐶) |
9 | 3, 5, 8 | vtoclbg 3473 | 1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∀wal 1541 = wceq 1543 ∈ wcel 2112 Ⅎwnfc 2877 ⦋csb 3798 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-ex 1788 df-nf 1792 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-v 3400 df-sbc 3684 df-csb 3799 |
This theorem is referenced by: cdlemefrs29bpre0 38096 |
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