Theorem riotasbc 7399

Description: Substitution law for descriptions. Compare iotasbc 45539. (Contributed by NM, 23-Aug-2011.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)

Assertion

Ref	Expression
riotasbc	⊢ (∃!𝑥 ∈ 𝐴 𝜑 → [(℩𝑥 ∈ 𝐴 𝜑) / 𝑥]𝜑)

Proof of Theorem riotasbc

Step	Hyp	Ref	Expression
1		rabssab 4009	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜑}
2		riotacl2 7397	. . 3 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) ∈ {𝑥 ∈ 𝐴 ∣ 𝜑})
3	1, 2	sselid 3907	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) ∈ {𝑥 ∣ 𝜑})
4		df-sbc 3739	. 2 ⊢ ([(℩𝑥 ∈ 𝐴 𝜑) / 𝑥]𝜑 ↔ (℩𝑥 ∈ 𝐴 𝜑) ∈ {𝑥 ∣ 𝜑})
5	3, 4	sylibr 237	1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → [(℩𝑥 ∈ 𝐴 𝜑) / 𝑥]𝜑)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2144  {cab 2738  ∃!wreu 3363  {crab 3412  [wsbc 3738  ℩crio 7380

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-12 2212  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  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-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-ss 3897  df-un 4150  df-sn 4585  df-pr 4587  df-uni 4867  df-iota 6497  df-riota 7381

This theorem is used by:  riotass2  7411  riotass  7412  cjth  15279  joinlem  18564  meetlem  18578  finxpreclem4  38454  poimirlem26  38701  riotasvd  40150  lshpkrlem3  40306  tfsconcatfv  44480