Theorem iotabi 5264

Description: Equivalence theorem for descriptions. (Contributed by Andrew Salmon, 30-Jun-2011.)

Assertion

Ref	Expression
iotabi	⊢ (∀𝑥(𝜑 ↔ 𝜓) → (℩𝑥𝜑) = (℩𝑥𝜓))

Proof of Theorem iotabi

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		abbi 2323	. . . . . 6 ⊢ (∀𝑥(𝜑 ↔ 𝜓) ↔ {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜓})
2	1	biimpi 120	. . . . 5 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜓})
3	2	eqeq1d 2218	. . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → ({𝑥 ∣ 𝜑} = {𝑧} ↔ {𝑥 ∣ 𝜓} = {𝑧}))
4	3	abbidv 2327	. . 3 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}} = {𝑧 ∣ {𝑥 ∣ 𝜓} = {𝑧}})
5	4	unieqd 3878	. 2 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → ∪ {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}} = ∪ {𝑧 ∣ {𝑥 ∣ 𝜓} = {𝑧}})
6		df-iota 5254	. 2 ⊢ (℩𝑥𝜑) = ∪ {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}}
7		df-iota 5254	. 2 ⊢ (℩𝑥𝜓) = ∪ {𝑧 ∣ {𝑥 ∣ 𝜓} = {𝑧}}
8	5, 6, 7	3eqtr4g 2267	1 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (℩𝑥𝜑) = (℩𝑥𝜓))

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1373   = wceq 1375  {cab 2195  {csn 3646  ∪ cuni 3867  ℩cio 5252

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-ext 2191

This theorem depends on definitions:  df-bi 117  df-tru 1378  df-nf 1487  df-sb 1789  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-rex 2494  df-uni 3868  df-iota 5254

This theorem is referenced by:  iotabidv  5277  iotabii  5278  iotaexel  5932  eusvobj1  5961