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Mirrors > Home > MPE Home > Th. List > abbi | Structured version Visualization version GIF version |
Description: Equivalent formulas define equal class abstractions, and conversely. (Contributed by NM, 25-Nov-2013.) (Revised by Mario Carneiro, 11-Aug-2016.) Remove dependency on ax-8 2113 and df-clel 2870 (by avoiding use of cleqh 2913). (Revised by BJ, 23-Jun-2019.) |
Ref | Expression |
---|---|
abbi | ⊢ (∀𝑥(𝜑 ↔ 𝜓) ↔ {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜓}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfcleq 2792 | . 2 ⊢ ({𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜓} ↔ ∀𝑦(𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝑦 ∈ {𝑥 ∣ 𝜓})) | |
2 | nfsab1 2785 | . . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∣ 𝜑} | |
3 | nfsab1 2785 | . . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∣ 𝜓} | |
4 | 2, 3 | nfbi 1904 | . . 3 ⊢ Ⅎ𝑥(𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝑦 ∈ {𝑥 ∣ 𝜓}) |
5 | nfv 1915 | . . 3 ⊢ Ⅎ𝑦(𝜑 ↔ 𝜓) | |
6 | df-clab 2777 | . . . . 5 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑) | |
7 | sbequ12r 2251 | . . . . 5 ⊢ (𝑦 = 𝑥 → ([𝑦 / 𝑥]𝜑 ↔ 𝜑)) | |
8 | 6, 7 | syl5bb 286 | . . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)) |
9 | df-clab 2777 | . . . . 5 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜓} ↔ [𝑦 / 𝑥]𝜓) | |
10 | sbequ12r 2251 | . . . . 5 ⊢ (𝑦 = 𝑥 → ([𝑦 / 𝑥]𝜓 ↔ 𝜓)) | |
11 | 9, 10 | syl5bb 286 | . . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ {𝑥 ∣ 𝜓} ↔ 𝜓)) |
12 | 8, 11 | bibi12d 349 | . . 3 ⊢ (𝑦 = 𝑥 → ((𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝑦 ∈ {𝑥 ∣ 𝜓}) ↔ (𝜑 ↔ 𝜓))) |
13 | 4, 5, 12 | cbvalv1 2350 | . 2 ⊢ (∀𝑦(𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝑦 ∈ {𝑥 ∣ 𝜓}) ↔ ∀𝑥(𝜑 ↔ 𝜓)) |
14 | 1, 13 | bitr2i 279 | 1 ⊢ (∀𝑥(𝜑 ↔ 𝜓) ↔ {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜓}) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∀wal 1536 = wceq 1538 [wsb 2069 ∈ wcel 2111 {cab 2776 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 |
This theorem is referenced by: nabbi 3089 rabbi 3336 absn 4543 karden 9308 elnev 41142 csbingVD 41590 csbsngVD 41599 csbxpgVD 41600 csbrngVD 41602 csbunigVD 41604 csbfv12gALTVD 41605 |
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