Theorem aiotaval 48259

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 48190	. . . . 5 ⊢ ∃!𝑧{𝑧} = {𝑦}
2		eqcom 2767	. . . . . 6 ⊢ ({𝑦} = {𝑧} ↔ {𝑧} = {𝑦})
3	2	eubii 2610	. . . . 5 ⊢ (∃!𝑧{𝑦} = {𝑧} ↔ ∃!𝑧{𝑧} = {𝑦})
4	1, 3	mpbir 234	. . . 4 ⊢ ∃!𝑧{𝑦} = {𝑧}
5		eqeq1 2764	. . . . 5 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → ({𝑥 ∣ 𝜑} = {𝑧} ↔ {𝑦} = {𝑧}))
6	5	eubidv 2611	. . . 4 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → (∃!𝑧{𝑥 ∣ 𝜑} = {𝑧} ↔ ∃!𝑧{𝑦} = {𝑧}))
7	4, 6	mpbiri 261	. . 3 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → ∃!𝑧{𝑥 ∣ 𝜑} = {𝑧})
8		absn 4600	. . 3 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} ↔ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
9		reuabaiotaiota 48251	. . . 4 ⊢ (∃!𝑧{𝑥 ∣ 𝜑} = {𝑧} ↔ (℩𝑥𝜑) = (℩'𝑥𝜑))
10		eqcom 2767	. . . 4 ⊢ ((℩𝑥𝜑) = (℩'𝑥𝜑) ↔ (℩'𝑥𝜑) = (℩𝑥𝜑))
11	9, 10	bitri 278	. . 3 ⊢ (∃!𝑧{𝑥 ∣ 𝜑} = {𝑧} ↔ (℩'𝑥𝜑) = (℩𝑥𝜑))
12	7, 8, 11	3imtr3i 294	. 2 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (℩'𝑥𝜑) = (℩𝑥𝜑))
13		iotaval 6512	. 2 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦)
14	12, 13	eqtrd 2795	1 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (℩'𝑥𝜑) = 𝑦)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1567   = wceq 1569  ∃!weu 2593  {cab 2738  {csn 4584  ℩cio 6492  ℩'caiota 48247

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-sn 4585  df-pr 4587  df-uni 4867  df-int 4907  df-iota 6494  df-aiota 48249

This theorem is used by:  aiota0def  48260