Theorem iotam 5209

Description: Representation of "the unique element such that 𝜑 " with a class expression 𝐴 which is inhabited (that means that "the unique element such that 𝜑 " exists). (Contributed by AV, 30-Jan-2024.)

Hypothesis

Ref	Expression
iotam.1	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
iotam	⊢ ((𝐴 ∈ 𝑉 ∧ ∃𝑤 𝑤 ∈ 𝐴 ∧ 𝐴 = (℩𝑥𝜑)) → 𝜓)

Distinct variable groups:   𝑥,𝐴   𝑤,𝐴   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑥,𝑤)   𝜓(𝑤)   𝑉(𝑥,𝑤)

Proof of Theorem iotam

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1w 2238	. . . . 5 ⊢ (𝑤 = 𝑧 → (𝑤 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
2	1	cbvexv 1918	. . . 4 ⊢ (∃𝑤 𝑤 ∈ 𝐴 ↔ ∃𝑧 𝑧 ∈ 𝐴)
3		simprr 531	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝐴 ∧ (𝐴 ∈ 𝑉 ∧ 𝐴 = (℩𝑥𝜑))) → 𝐴 = (℩𝑥𝜑))
4	3	eqcomd 2183	. . . . . . 7 ⊢ ((𝑧 ∈ 𝐴 ∧ (𝐴 ∈ 𝑉 ∧ 𝐴 = (℩𝑥𝜑))) → (℩𝑥𝜑) = 𝐴)
5		simprl 529	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝐴 ∧ (𝐴 ∈ 𝑉 ∧ 𝐴 = (℩𝑥𝜑))) → 𝐴 ∈ 𝑉)
6		simpl 109	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝐴 ∧ (𝐴 ∈ 𝑉 ∧ 𝐴 = (℩𝑥𝜑))) → 𝑧 ∈ 𝐴)
7	6, 3	eleqtrd 2256	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝐴 ∧ (𝐴 ∈ 𝑉 ∧ 𝐴 = (℩𝑥𝜑))) → 𝑧 ∈ (℩𝑥𝜑))
8		eliotaeu 5206	. . . . . . . . 9 ⊢ (𝑧 ∈ (℩𝑥𝜑) → ∃!𝑥𝜑)
9	7, 8	syl 14	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝐴 ∧ (𝐴 ∈ 𝑉 ∧ 𝐴 = (℩𝑥𝜑))) → ∃!𝑥𝜑)
10		iotam.1	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
11	10	iota2 5207	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ ∃!𝑥𝜑) → (𝜓 ↔ (℩𝑥𝜑) = 𝐴))
12	5, 9, 11	syl2anc 411	. . . . . . 7 ⊢ ((𝑧 ∈ 𝐴 ∧ (𝐴 ∈ 𝑉 ∧ 𝐴 = (℩𝑥𝜑))) → (𝜓 ↔ (℩𝑥𝜑) = 𝐴))
13	4, 12	mpbird 167	. . . . . 6 ⊢ ((𝑧 ∈ 𝐴 ∧ (𝐴 ∈ 𝑉 ∧ 𝐴 = (℩𝑥𝜑))) → 𝜓)
14	13	ex 115	. . . . 5 ⊢ (𝑧 ∈ 𝐴 → ((𝐴 ∈ 𝑉 ∧ 𝐴 = (℩𝑥𝜑)) → 𝜓))
15	14	exlimiv 1598	. . . 4 ⊢ (∃𝑧 𝑧 ∈ 𝐴 → ((𝐴 ∈ 𝑉 ∧ 𝐴 = (℩𝑥𝜑)) → 𝜓))
16	2, 15	sylbi 121	. . 3 ⊢ (∃𝑤 𝑤 ∈ 𝐴 → ((𝐴 ∈ 𝑉 ∧ 𝐴 = (℩𝑥𝜑)) → 𝜓))
17	16	3impib 1201	. 2 ⊢ ((∃𝑤 𝑤 ∈ 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝐴 = (℩𝑥𝜑)) → 𝜓)
18	17	3com12 1207	1 ⊢ ((𝐴 ∈ 𝑉 ∧ ∃𝑤 𝑤 ∈ 𝐴 ∧ 𝐴 = (℩𝑥𝜑)) → 𝜓)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 978   = wceq 1353  ∃wex 1492  ∃!weu 2026   ∈ wcel 2148  ℩cio 5177

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-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2740  df-sbc 2964  df-un 3134  df-sn 3599  df-pr 3600  df-uni 3811  df-iota 5179

This theorem is referenced by:  sgrpidmndm  12821