Theorem iotaeq 5181

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

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 1799	. . . . . . 7 ⊢ (∀𝑥 𝑥 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜑))
2		df-clab 2164	. . . . . . 7 ⊢ (𝑧 ∈ {𝑥 ∣ 𝜑} ↔ [𝑧 / 𝑥]𝜑)
3		df-clab 2164	. . . . . . 7 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜑} ↔ [𝑧 / 𝑦]𝜑)
4	1, 2, 3	3bitr4g 223	. . . . . 6 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝑧 ∈ {𝑥 ∣ 𝜑} ↔ 𝑧 ∈ {𝑦 ∣ 𝜑}))
5	4	eqrdv 2175	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜑})
6	5	eqeq1d 2186	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → ({𝑥 ∣ 𝜑} = {𝑧} ↔ {𝑦 ∣ 𝜑} = {𝑧}))
7	6	abbidv 2295	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}} = {𝑧 ∣ {𝑦 ∣ 𝜑} = {𝑧}})
8	7	unieqd 3818	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∪ {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}} = ∪ {𝑧 ∣ {𝑦 ∣ 𝜑} = {𝑧}})
9		df-iota 5173	. 2 ⊢ (℩𝑥𝜑) = ∪ {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}}
10		df-iota 5173	. 2 ⊢ (℩𝑦𝜑) = ∪ {𝑧 ∣ {𝑦 ∣ 𝜑} = {𝑧}}
11	8, 9, 10	3eqtr4g 2235	1 ⊢ (∀𝑥 𝑥 = 𝑦 → (℩𝑥𝜑) = (℩𝑦𝜑))

Syntax hints:   → wi 4  ∀wal 1351   = wceq 1353  [wsb 1762   ∈ wcel 2148  {cab 2163  {csn 3591  ∪ cuni 3807  ℩cio 5171

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-uni 3808  df-iota 5173

This theorem is referenced by: (None)