Theorem aiotaval 47010

Description: Theorem 8.19 in [Quine] p. 57. This theorem is the fundamental property of (alternate) iota. (Contributed by AV, 24-Aug-2022.)

Assertion

Ref	Expression
aiotaval	⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (℩'𝑥𝜑) = 𝑦)

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem aiotaval

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eusnsn 46941	. . . . 5 ⊢ ∃!𝑧{𝑧} = {𝑦}
2		eqcom 2747	. . . . . 6 ⊢ ({𝑦} = {𝑧} ↔ {𝑧} = {𝑦})
3	2	eubii 2588	. . . . 5 ⊢ (∃!𝑧{𝑦} = {𝑧} ↔ ∃!𝑧{𝑧} = {𝑦})
4	1, 3	mpbir 231	. . . 4 ⊢ ∃!𝑧{𝑦} = {𝑧}
5		eqeq1 2744	. . . . 5 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → ({𝑥 ∣ 𝜑} = {𝑧} ↔ {𝑦} = {𝑧}))
6	5	eubidv 2589	. . . 4 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → (∃!𝑧{𝑥 ∣ 𝜑} = {𝑧} ↔ ∃!𝑧{𝑦} = {𝑧}))
7	4, 6	mpbiri 258	. . 3 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → ∃!𝑧{𝑥 ∣ 𝜑} = {𝑧})
8		absn 4667	. . 3 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} ↔ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
9		reuabaiotaiota 47002	. . . 4 ⊢ (∃!𝑧{𝑥 ∣ 𝜑} = {𝑧} ↔ (℩𝑥𝜑) = (℩'𝑥𝜑))
10		eqcom 2747	. . . 4 ⊢ ((℩𝑥𝜑) = (℩'𝑥𝜑) ↔ (℩'𝑥𝜑) = (℩𝑥𝜑))
11	9, 10	bitri 275	. . 3 ⊢ (∃!𝑧{𝑥 ∣ 𝜑} = {𝑧} ↔ (℩'𝑥𝜑) = (℩𝑥𝜑))
12	7, 8, 11	3imtr3i 291	. 2 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (℩'𝑥𝜑) = (℩𝑥𝜑))
13		iotaval 6544	. 2 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦)
14	12, 13	eqtrd 2780	1 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (℩'𝑥𝜑) = 𝑦)

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1535   = wceq 1537  ∃!weu 2571  {cab 2717  {csn 4648  ℩cio 6523  ℩'caiota 46998

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-sn 4649  df-pr 4651  df-uni 4932  df-int 4971  df-iota 6525  df-aiota 47000

This theorem is referenced by:  aiota0def  47011