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Mirrors > Home > MPE Home > Th. List > iotabi | Structured version Visualization version 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 2902 | . . . . . 6 ⊢ (∀𝑥(𝜑 ↔ 𝜓) ↔ {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜓}) | |
2 | 1 | biimpi 208 | . . . . 5 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜓}) |
3 | 2 | eqeq1d 2780 | . . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → ({𝑥 ∣ 𝜑} = {𝑧} ↔ {𝑥 ∣ 𝜓} = {𝑧})) |
4 | 3 | abbidv 2906 | . . 3 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}} = {𝑧 ∣ {𝑥 ∣ 𝜓} = {𝑧}}) |
5 | 4 | unieqd 4681 | . 2 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → ∪ {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}} = ∪ {𝑧 ∣ {𝑥 ∣ 𝜓} = {𝑧}}) |
6 | df-iota 6099 | . 2 ⊢ (℩𝑥𝜑) = ∪ {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}} | |
7 | df-iota 6099 | . 2 ⊢ (℩𝑥𝜓) = ∪ {𝑧 ∣ {𝑥 ∣ 𝜓} = {𝑧}} | |
8 | 5, 6, 7 | 3eqtr4g 2839 | 1 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (℩𝑥𝜑) = (℩𝑥𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∀wal 1599 = wceq 1601 {cab 2763 {csn 4398 ∪ cuni 4671 ℩cio 6097 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-ext 2754 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-clab 2764 df-cleq 2770 df-clel 2774 df-rex 3096 df-uni 4672 df-iota 6099 |
This theorem is referenced by: iotabidv 6120 iotabii 6121 eusvobj1 6916 iotasbcq 39597 |
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