Theorem aiotaval 46102

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 46035	. . . . 5 ⊢ ∃!𝑧{𝑧} = {𝑦}
2		eqcom 2738	. . . . . 6 ⊢ ({𝑦} = {𝑧} ↔ {𝑧} = {𝑦})
3	2	eubii 2578	. . . . 5 ⊢ (∃!𝑧{𝑦} = {𝑧} ↔ ∃!𝑧{𝑧} = {𝑦})
4	1, 3	mpbir 230	. . . 4 ⊢ ∃!𝑧{𝑦} = {𝑧}
5		eqeq1 2735	. . . . 5 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → ({𝑥 ∣ 𝜑} = {𝑧} ↔ {𝑦} = {𝑧}))
6	5	eubidv 2579	. . . 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 4646	. . 3 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} ↔ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
9		reuabaiotaiota 46094	. . . 4 ⊢ (∃!𝑧{𝑥 ∣ 𝜑} = {𝑧} ↔ (℩𝑥𝜑) = (℩'𝑥𝜑))
10		eqcom 2738	. . . 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 6514	. 2 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦)
14	12, 13	eqtrd 2771	1 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (℩'𝑥𝜑) = 𝑦)

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1538   = wceq 1540  ∃!weu 2561  {cab 2708  {csn 4628  ℩cio 6493  ℩'caiota 46090

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-11 2153  ax-12 2170  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-sn 4629  df-pr 4631  df-uni 4909  df-int 4951  df-iota 6495  df-aiota 46092

This theorem is referenced by:  aiota0def  46103