Theorem iotabi 5097

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 2253	. . . . . 6 ⊢ (∀𝑥(𝜑 ↔ 𝜓) ↔ {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜓})
2	1	biimpi 119	. . . . 5 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜓})
3	2	eqeq1d 2148	. . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → ({𝑥 ∣ 𝜑} = {𝑧} ↔ {𝑥 ∣ 𝜓} = {𝑧}))
4	3	abbidv 2257	. . 3 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}} = {𝑧 ∣ {𝑥 ∣ 𝜓} = {𝑧}})
5	4	unieqd 3747	. 2 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → ∪ {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}} = ∪ {𝑧 ∣ {𝑥 ∣ 𝜓} = {𝑧}})
6		df-iota 5088	. 2 ⊢ (℩𝑥𝜑) = ∪ {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}}
7		df-iota 5088	. 2 ⊢ (℩𝑥𝜓) = ∪ {𝑧 ∣ {𝑥 ∣ 𝜓} = {𝑧}}
8	5, 6, 7	3eqtr4g 2197	1 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (℩𝑥𝜑) = (℩𝑥𝜓))

Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1329   = wceq 1331  {cab 2125  {csn 3527  ∪ cuni 3736  ℩cio 5086

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rex 2422  df-uni 3737  df-iota 5088

This theorem is referenced by:  iotabidv  5109  iotabii  5110  eusvobj1  5761