Theorem iotaeq 5054

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 1753	. . . . . . 7 ⊢ (∀𝑥 𝑥 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜑))
2		df-clab 2102	. . . . . . 7 ⊢ (𝑧 ∈ {𝑥 ∣ 𝜑} ↔ [𝑧 / 𝑥]𝜑)
3		df-clab 2102	. . . . . . 7 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜑} ↔ [𝑧 / 𝑦]𝜑)
4	1, 2, 3	3bitr4g 222	. . . . . 6 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝑧 ∈ {𝑥 ∣ 𝜑} ↔ 𝑧 ∈ {𝑦 ∣ 𝜑}))
5	4	eqrdv 2113	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜑})
6	5	eqeq1d 2123	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → ({𝑥 ∣ 𝜑} = {𝑧} ↔ {𝑦 ∣ 𝜑} = {𝑧}))
7	6	abbidv 2232	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}} = {𝑧 ∣ {𝑦 ∣ 𝜑} = {𝑧}})
8	7	unieqd 3713	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∪ {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}} = ∪ {𝑧 ∣ {𝑦 ∣ 𝜑} = {𝑧}})
9		df-iota 5046	. 2 ⊢ (℩𝑥𝜑) = ∪ {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}}
10		df-iota 5046	. 2 ⊢ (℩𝑦𝜑) = ∪ {𝑧 ∣ {𝑦 ∣ 𝜑} = {𝑧}}
11	8, 9, 10	3eqtr4g 2172	1 ⊢ (∀𝑥 𝑥 = 𝑦 → (℩𝑥𝜑) = (℩𝑦𝜑))

Syntax hints:   → wi 4  ∀wal 1312   = wceq 1314   ∈ wcel 1463  [wsb 1718  {cab 2101  {csn 3493  ∪ cuni 3702  ℩cio 5044

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097

This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-rex 2396  df-uni 3703  df-iota 5046

This theorem is referenced by: (None)