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Mirrors > Home > ILE Home > Th. List > iotabi | GIF version |
Description: Equivalence theorem for descriptions. (Contributed by Andrew Salmon, 30-Jun-2011.) |
Ref | Expression |
---|---|
iotabi | ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (℩𝑥𝜑) = (℩𝑥𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abbi 2254 | . . . . . 6 ⊢ (∀𝑥(𝜑 ↔ 𝜓) ↔ {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜓}) | |
2 | 1 | biimpi 119 | . . . . 5 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜓}) |
3 | 2 | eqeq1d 2149 | . . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → ({𝑥 ∣ 𝜑} = {𝑧} ↔ {𝑥 ∣ 𝜓} = {𝑧})) |
4 | 3 | abbidv 2258 | . . 3 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}} = {𝑧 ∣ {𝑥 ∣ 𝜓} = {𝑧}}) |
5 | 4 | unieqd 3755 | . 2 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → ∪ {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}} = ∪ {𝑧 ∣ {𝑥 ∣ 𝜓} = {𝑧}}) |
6 | df-iota 5096 | . 2 ⊢ (℩𝑥𝜑) = ∪ {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}} | |
7 | df-iota 5096 | . 2 ⊢ (℩𝑥𝜓) = ∪ {𝑧 ∣ {𝑥 ∣ 𝜓} = {𝑧}} | |
8 | 5, 6, 7 | 3eqtr4g 2198 | 1 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (℩𝑥𝜑) = (℩𝑥𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∀wal 1330 = wceq 1332 {cab 2126 {csn 3532 ∪ cuni 3744 ℩cio 5094 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-rex 2423 df-uni 3745 df-iota 5096 |
This theorem is referenced by: iotabidv 5117 iotabii 5118 eusvobj1 5769 |
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