Theorem riotasbc 6626

Description: Substitution law for descriptions. Compare iotasbc 38620. (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 3690	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜑}
2		riotacl2 6624	. . 3 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) ∈ {𝑥 ∈ 𝐴 ∣ 𝜑})
3	1, 2	sseldi 3601	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) ∈ {𝑥 ∣ 𝜑})
4		df-sbc 3436	. 2 ⊢ ([(℩𝑥 ∈ 𝐴 𝜑) / 𝑥]𝜑 ↔ (℩𝑥 ∈ 𝐴 𝜑) ∈ {𝑥 ∣ 𝜑})
5	3, 4	sylibr 224	1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → [(℩𝑥 ∈ 𝐴 𝜑) / 𝑥]𝜑)

Syntax hints:   → wi 4   ∈ wcel 1990  {cab 2608  ∃!wreu 2914  {crab 2916  [wsbc 3435  ℩crio 6610

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-un 3579  df-in 3581  df-ss 3588  df-sn 4178  df-pr 4180  df-uni 4437  df-iota 5851  df-riota 6611

This theorem is referenced by:  riotass2  6638  riotass  6639  cjth  13843  joinlem  17011  meetlem  17025  finxpreclem4  33231  poimirlem26  33435  riotasvd  34242  lshpkrlem3  34399