Theorem riotasbc 5407

Description: Substitution law for descriptions. (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 3004	. . 3 ⊢ {x ∈ A ∣ φ} ⊆ {x ∣ φ}
2		riotacl2 5405	. . 3 ⊢ (∃!x ∈ A φ → (℩x ∈ A φ) ∈ {x ∈ A ∣ φ})
3	1, 2	sseldi 2920	. 2 ⊢ (∃!x ∈ A φ → (℩x ∈ A φ) ∈ {x ∣ φ})
4		df-sbc 2742	. 2 ⊢ ([(℩x ∈ A φ) / x]φ ↔ (℩x ∈ A φ) ∈ {x ∣ φ})
5	3, 4	sylibr 137	1 ⊢ (∃!x ∈ A φ → [(℩x ∈ A φ) / x]φ)

Syntax hints:   → wi 4   ∈ wcel 1374  {cab 2008  ∃!wreu 2286  {crab 2288  [wsbc 2741  ℩crio 5392

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004

This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-rex 2290  df-reu 2291  df-rab 2293  df-v 2537  df-sbc 2742  df-un 2899  df-in 2901  df-ss 2908  df-sn 3356  df-pr 3357  df-uni 3555  df-iota 4794  df-riota 5393

This theorem is referenced by:  riotass2  5418  riotass  5419