Theorem riotasbc 7386

Description: Substitution law for descriptions. Compare iotasbc 43480. (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 4082	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜑}
2		riotacl2 7384	. . 3 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) ∈ {𝑥 ∈ 𝐴 ∣ 𝜑})
3	1, 2	sselid 3979	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) ∈ {𝑥 ∣ 𝜑})
4		df-sbc 3777	. 2 ⊢ ([(℩𝑥 ∈ 𝐴 𝜑) / 𝑥]𝜑 ↔ (℩𝑥 ∈ 𝐴 𝜑) ∈ {𝑥 ∣ 𝜑})
5	3, 4	sylibr 233	1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → [(℩𝑥 ∈ 𝐴 𝜑) / 𝑥]𝜑)

Syntax hints:   → wi 4   ∈ wcel 2104  {cab 2707  ∃!wreu 3372  {crab 3430  [wsbc 3776  ℩crio 7366

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-12 2169  ax-ext 2701

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-tru 1542  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-un 3952  df-in 3954  df-ss 3964  df-sn 4628  df-pr 4630  df-uni 4908  df-iota 6494  df-riota 7367

This theorem is referenced by:  riotass2  7398  riotass  7399  cjth  15054  joinlem  18340  meetlem  18354  finxpreclem4  36578  poimirlem26  36817  riotasvd  38129  lshpkrlem3  38285  tfsconcatfv  42393