Theorem riotasbc 7423

Description: Substitution law for descriptions. Compare iotasbc 44388. (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 4108	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜑}
2		riotacl2 7421	. . 3 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) ∈ {𝑥 ∈ 𝐴 ∣ 𝜑})
3	1, 2	sselid 4006	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) ∈ {𝑥 ∣ 𝜑})
4		df-sbc 3805	. 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 2108  {cab 2717  ∃!wreu 3386  {crab 3443  [wsbc 3804  ℩crio 7403

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2178  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-un 3981  df-ss 3993  df-sn 4649  df-pr 4651  df-uni 4932  df-iota 6525  df-riota 7404

This theorem is referenced by:  riotass2  7435  riotass  7436  cjth  15152  joinlem  18453  meetlem  18467  finxpreclem4  37360  poimirlem26  37606  riotasvd  38912  lshpkrlem3  39068  tfsconcatfv  43303