Theorem iotaeq 6508

Description: Equality theorem for descriptions. Usage of this theorem is discouraged because it depends on ax-13 2370. (Contributed by Andrew Salmon, 30-Jun-2011.) (New usage is discouraged.)

Assertion

Ref	Expression
iotaeq	⊢ (∀𝑥 𝑥 = 𝑦 → (℩𝑥𝜑) = (℩𝑦𝜑))

Proof of Theorem iotaeq

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		drsb1 2493	. . . . . . 7 ⊢ (∀𝑥 𝑥 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜑))
2		df-clab 2709	. . . . . . 7 ⊢ (𝑧 ∈ {𝑥 ∣ 𝜑} ↔ [𝑧 / 𝑥]𝜑)
3		df-clab 2709	. . . . . . 7 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜑} ↔ [𝑧 / 𝑦]𝜑)
4	1, 2, 3	3bitr4g 314	. . . . . 6 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝑧 ∈ {𝑥 ∣ 𝜑} ↔ 𝑧 ∈ {𝑦 ∣ 𝜑}))
5	4	eqrdv 2729	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜑})
6	5	eqeq1d 2733	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → ({𝑥 ∣ 𝜑} = {𝑧} ↔ {𝑦 ∣ 𝜑} = {𝑧}))
7	6	abbidv 2800	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}} = {𝑧 ∣ {𝑦 ∣ 𝜑} = {𝑧}})
8	7	unieqd 4922	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∪ {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}} = ∪ {𝑧 ∣ {𝑦 ∣ 𝜑} = {𝑧}})
9		df-iota 6495	. 2 ⊢ (℩𝑥𝜑) = ∪ {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}}
10		df-iota 6495	. 2 ⊢ (℩𝑦𝜑) = ∪ {𝑧 ∣ {𝑦 ∣ 𝜑} = {𝑧}}
11	8, 9, 10	3eqtr4g 2796	1 ⊢ (∀𝑥 𝑥 = 𝑦 → (℩𝑥𝜑) = (℩𝑦𝜑))

Syntax hints:   → wi 4  ∀wal 1538   = wceq 1540  [wsb 2066   ∈ wcel 2105  {cab 2708  {csn 4628  ∪ cuni 4908  ℩cio 6493

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-12 2170  ax-13 2370  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-v 3475  df-in 3955  df-ss 3965  df-uni 4909  df-iota 6495

This theorem is referenced by: (None)