Theorem iotaval 6502

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 2141, ax-11 2157, ax-12 2177. (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 2800	. . 3 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝑥 = 𝑦})
2		df-sn 4602	. . 3 ⊢ {𝑦} = {𝑥 ∣ 𝑥 = 𝑦}
3	1, 2	eqtr4di 2788	. 2 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → {𝑥 ∣ 𝜑} = {𝑦})
4		iotaval2 6499	. 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 2713  {csn 4601  ℩cio 6482

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 2007  ax-8 2110  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-v 3461  df-un 3931  df-ss 3943  df-sn 4602  df-pr 4604  df-uni 4884  df-iota 6484

This theorem is referenced by:  iotauni  6506  iota1  6508  iotaexOLD  6511  iota4  6512  iota5  6514  iota5f  35741  iotain  44441  iotaexeu  44442  iotasbc  44443  iotaequ  44453  iotavalb  44454  pm14.24  44456  sbiota1  44458  aiotaval  47124