Theorem riotasbc 7344

Description: Substitution law for descriptions. Compare iotasbc 44381. (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 4044	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜑}
2		riotacl2 7342	. . 3 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) ∈ {𝑥 ∈ 𝐴 ∣ 𝜑})
3	1, 2	sselid 3941	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) ∈ {𝑥 ∣ 𝜑})
4		df-sbc 3751	. 2 ⊢ ([(℩𝑥 ∈ 𝐴 𝜑) / 𝑥]𝜑 ↔ (℩𝑥 ∈ 𝐴 𝜑) ∈ {𝑥 ∣ 𝜑})
5	3, 4	sylibr 234	1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → [(℩𝑥 ∈ 𝐴 𝜑) / 𝑥]𝜑)

Syntax hints:   → wi 4   ∈ wcel 2109  {cab 2707  ∃!wreu 3349  {crab 3402  [wsbc 3750  ℩crio 7325

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-10 2142  ax-12 2178  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-un 3916  df-ss 3928  df-sn 4586  df-pr 4588  df-uni 4868  df-iota 6452  df-riota 7326

This theorem is referenced by:  riotass2  7356  riotass  7357  cjth  15045  joinlem  18318  meetlem  18332  finxpreclem4  37355  poimirlem26  37613  riotasvd  38922  lshpkrlem3  39078  tfsconcatfv  43303