Theorem iotabi 6500

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 2826	. . . . 5 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜓})
2	1	eqeq1d 2763	. . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → ({𝑥 ∣ 𝜑} = {𝑧} ↔ {𝑥 ∣ 𝜓} = {𝑧}))
3	2	abbidv 2827	. . 3 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}} = {𝑧 ∣ {𝑥 ∣ 𝜓} = {𝑧}})
4	3	unieqd 4880	. 2 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → ∪ {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}} = ∪ {𝑧 ∣ {𝑥 ∣ 𝜓} = {𝑧}})
5		df-iota 6487	. 2 ⊢ (℩𝑥𝜑) = ∪ {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}}
6		df-iota 6487	. 2 ⊢ (℩𝑥𝜓) = ∪ {𝑧 ∣ {𝑥 ∣ 𝜓} = {𝑧}}
7	4, 5, 6	3eqtr4g 2821	1 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (℩𝑥𝜑) = (℩𝑥𝜓))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1568   = wceq 1570  {cab 2739  {csn 4584  ∪ cuni 4867  ℩cio 6485

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3453  df-ss 3916  df-uni 4868  df-iota 6487

This theorem is used by:  iotabidv  6515  iotabii  6516  eusvobj1  7405  iotasbcq  45379