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Mirrors > Home > MPE Home > Th. List > abbib | Structured version Visualization version GIF version |
Description: Equal class abstractions require equivalent formulas, and conversely. (Contributed by NM, 25-Nov-2013.) (Revised by Mario Carneiro, 11-Aug-2016.) Remove dependency on ax-8 2108 and df-clel 2814 (by avoiding use of cleqh 2869). (Revised by BJ, 23-Jun-2019.) Definitial form. (Revised by Wolf Lammen, 23-Feb-2025.) |
Ref | Expression |
---|---|
abbib | ⊢ ({𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜓} ↔ ∀𝑥(𝜑 ↔ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfcleq 2728 | . 2 ⊢ ({𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜓} ↔ ∀𝑦(𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝑦 ∈ {𝑥 ∣ 𝜓})) | |
2 | nfsab1 2720 | . . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∣ 𝜑} | |
3 | nfsab1 2720 | . . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∣ 𝜓} | |
4 | 2, 3 | nfbi 1901 | . . 3 ⊢ Ⅎ𝑥(𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝑦 ∈ {𝑥 ∣ 𝜓}) |
5 | nfv 1912 | . . 3 ⊢ Ⅎ𝑦(𝜑 ↔ 𝜓) | |
6 | df-clab 2713 | . . . . 5 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑) | |
7 | sbequ12r 2250 | . . . . 5 ⊢ (𝑦 = 𝑥 → ([𝑦 / 𝑥]𝜑 ↔ 𝜑)) | |
8 | 6, 7 | bitrid 283 | . . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)) |
9 | df-clab 2713 | . . . . 5 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜓} ↔ [𝑦 / 𝑥]𝜓) | |
10 | sbequ12r 2250 | . . . . 5 ⊢ (𝑦 = 𝑥 → ([𝑦 / 𝑥]𝜓 ↔ 𝜓)) | |
11 | 9, 10 | bitrid 283 | . . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ {𝑥 ∣ 𝜓} ↔ 𝜓)) |
12 | 8, 11 | bibi12d 345 | . . 3 ⊢ (𝑦 = 𝑥 → ((𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝑦 ∈ {𝑥 ∣ 𝜓}) ↔ (𝜑 ↔ 𝜓))) |
13 | 4, 5, 12 | cbvalv1 2342 | . 2 ⊢ (∀𝑦(𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝑦 ∈ {𝑥 ∣ 𝜓}) ↔ ∀𝑥(𝜑 ↔ 𝜓)) |
14 | 1, 13 | bitri 275 | 1 ⊢ ({𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜓} ↔ ∀𝑥(𝜑 ↔ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∀wal 1535 = wceq 1537 [wsb 2062 ∈ wcel 2106 {cab 2712 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 |
This theorem is referenced by: eqabb 2879 nabbib 3043 rabbi 3465 ab0 4386 absn 4650 karden 9933 abeqabi 43398 elnev 44434 csbingVD 44882 csbsngVD 44891 csbxpgVD 44892 csbrngVD 44894 csbunigVD 44896 csbfv12gALTVD 44897 |
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