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Mirrors > Home > ILE Home > Th. List > abbi | GIF version |
Description: Equivalent wff's correspond to equal class abstractions. (Contributed by NM, 25-Nov-2013.) (Revised by Mario Carneiro, 11-Aug-2016.) |
Ref | Expression |
---|---|
abbi | ⊢ (∀x(φ ↔ ψ) ↔ {x ∣ φ} = {x ∣ ψ}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfcleq 2031 | . 2 ⊢ ({x ∣ φ} = {x ∣ ψ} ↔ ∀y(y ∈ {x ∣ φ} ↔ y ∈ {x ∣ ψ})) | |
2 | nfsab1 2027 | . . . 4 ⊢ Ⅎx y ∈ {x ∣ φ} | |
3 | nfsab1 2027 | . . . 4 ⊢ Ⅎx y ∈ {x ∣ ψ} | |
4 | 2, 3 | nfbi 1478 | . . 3 ⊢ Ⅎx(y ∈ {x ∣ φ} ↔ y ∈ {x ∣ ψ}) |
5 | nfv 1418 | . . 3 ⊢ Ⅎy(φ ↔ ψ) | |
6 | df-clab 2024 | . . . . 5 ⊢ (y ∈ {x ∣ φ} ↔ [y / x]φ) | |
7 | sbequ12r 1652 | . . . . 5 ⊢ (y = x → ([y / x]φ ↔ φ)) | |
8 | 6, 7 | syl5bb 181 | . . . 4 ⊢ (y = x → (y ∈ {x ∣ φ} ↔ φ)) |
9 | df-clab 2024 | . . . . 5 ⊢ (y ∈ {x ∣ ψ} ↔ [y / x]ψ) | |
10 | sbequ12r 1652 | . . . . 5 ⊢ (y = x → ([y / x]ψ ↔ ψ)) | |
11 | 9, 10 | syl5bb 181 | . . . 4 ⊢ (y = x → (y ∈ {x ∣ ψ} ↔ ψ)) |
12 | 8, 11 | bibi12d 224 | . . 3 ⊢ (y = x → ((y ∈ {x ∣ φ} ↔ y ∈ {x ∣ ψ}) ↔ (φ ↔ ψ))) |
13 | 4, 5, 12 | cbval 1634 | . 2 ⊢ (∀y(y ∈ {x ∣ φ} ↔ y ∈ {x ∣ ψ}) ↔ ∀x(φ ↔ ψ)) |
14 | 1, 13 | bitr2i 174 | 1 ⊢ (∀x(φ ↔ ψ) ↔ {x ∣ φ} = {x ∣ ψ}) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 98 ∀wal 1240 = wceq 1242 ∈ wcel 1390 [wsb 1642 {cab 2023 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-11 1394 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 |
This theorem is referenced by: abbii 2150 abbid 2151 rabbi 2481 dfiota2 4811 iotabi 4819 uniabio 4820 |
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