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Mirrors > Home > MPE Home > Th. List > abbi | Structured version Visualization version GIF version |
Description: Equivalent wff's correspond to equal class abstractions. (Contributed by NM, 25-Nov-2013.) (Revised by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 16-Nov-2019.) |
Ref | Expression |
---|---|
abbi | ⊢ (∀𝑥(𝜑 ↔ 𝜓) ↔ {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜓}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hbab1 2611 | . . 3 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} → ∀𝑥 𝑦 ∈ {𝑥 ∣ 𝜑}) | |
2 | hbab1 2611 | . . 3 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜓} → ∀𝑥 𝑦 ∈ {𝑥 ∣ 𝜓}) | |
3 | 1, 2 | cleqh 2724 | . 2 ⊢ ({𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜓} ↔ ∀𝑥(𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ {𝑥 ∣ 𝜓})) |
4 | abid 2610 | . . . 4 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑) | |
5 | abid 2610 | . . . 4 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜓} ↔ 𝜓) | |
6 | 4, 5 | bibi12i 329 | . . 3 ⊢ ((𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ {𝑥 ∣ 𝜓}) ↔ (𝜑 ↔ 𝜓)) |
7 | 6 | albii 1747 | . 2 ⊢ (∀𝑥(𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ {𝑥 ∣ 𝜓}) ↔ ∀𝑥(𝜑 ↔ 𝜓)) |
8 | 3, 7 | bitr2i 265 | 1 ⊢ (∀𝑥(𝜑 ↔ 𝜓) ↔ {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜓}) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∀wal 1481 = wceq 1483 ∈ wcel 1990 {cab 2608 |
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-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 |
This theorem is referenced by: abbii 2739 abbid 2740 nabbi 2896 rabbi 3120 sbcbi2 3484 rabeqsn 4214 iuneq12df 4544 dfiota2 5852 iotabi 5860 uniabio 5861 iotanul 5866 karden 8758 iuneq12daf 29373 bj-cleq 32949 abeq12 33964 elnev 38639 csbingVD 39120 csbsngVD 39129 csbxpgVD 39130 csbrngVD 39132 csbunigVD 39134 csbfv12gALTVD 39135 |
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