Theorem iotaval 6485

Description: Theorem 8.19 in [Quine] p. 57. This theorem is the fundamental property of iota. (Contributed by Andrew Salmon, 11-Jul-2011.) Remove dependency on ax-10 2142, ax-11 2158, ax-12 2178. (Revised by SN, 23-Nov-2024.)

Assertion

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

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem iotaval

Step	Hyp	Ref	Expression
1		abbi 2795	. . 3 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝑥 = 𝑦})
2		df-sn 4593	. . 3 ⊢ {𝑦} = {𝑥 ∣ 𝑥 = 𝑦}
3	1, 2	eqtr4di 2783	. 2 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → {𝑥 ∣ 𝜑} = {𝑦})
4		iotaval2 6482	. 2 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = 𝑦)
5	3, 4	syl 17	1 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦)

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1538   = wceq 1540  {cab 2708  {csn 4592  ℩cio 6465

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-v 3452  df-un 3922  df-ss 3934  df-sn 4593  df-pr 4595  df-uni 4875  df-iota 6467

This theorem is referenced by:  iotauni  6489  iota1  6491  iotaexOLD  6494  iota4  6495  iota5  6497  iota5f  35718  iotain  44413  iotaexeu  44414  iotasbc  44415  iotaequ  44425  iotavalb  44426  pm14.24  44428  sbiota1  44430  aiotaval  47100