Theorem riotaeqbidv 7235

Description: Equality deduction for restricted universal quantifier. (Contributed by NM, 15-Sep-2011.)

Hypotheses

Ref	Expression
riotaeqbidv.1	⊢ (𝜑 → 𝐴 = 𝐵)
riotaeqbidv.2	⊢ (𝜑 → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
riotaeqbidv	⊢ (𝜑 → (℩𝑥 ∈ 𝐴 𝜓) = (℩𝑥 ∈ 𝐵 𝜒))

Distinct variable group:   𝜑,𝑥

Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem riotaeqbidv

Step	Hyp	Ref	Expression
1		riotaeqbidv.2	. . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
2	1	riotabidv 7234	. 2 ⊢ (𝜑 → (℩𝑥 ∈ 𝐴 𝜓) = (℩𝑥 ∈ 𝐴 𝜒))
3		riotaeqbidv.1	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
4	3	riotaeqdv 7233	. 2 ⊢ (𝜑 → (℩𝑥 ∈ 𝐴 𝜒) = (℩𝑥 ∈ 𝐵 𝜒))
5	2, 4	eqtrd 2778	1 ⊢ (𝜑 → (℩𝑥 ∈ 𝐴 𝜓) = (℩𝑥 ∈ 𝐵 𝜒))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1539  ℩crio 7231

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-in 3894  df-ss 3904  df-uni 4840  df-iota 6391  df-riota 7232

This theorem is referenced by:  dfoi  9270  oieq1  9271  oieq2  9272  ordtypecbv  9276  ordtypelem3  9279  zorn2lem1  10252  zorn2g  10259  cidfval  17385  cidval  17386  cidpropd  17419  lubfval  18068  glbfval  18081  grpinvfval  18618  grpinvfvalALT  18619  pj1fval  19300  mpfrcl  21295  evlsval  21296  q1pval  25318  ig1pval  25337  mirval  27016  midf  27137  ismidb  27139  lmif  27146  islmib  27148  gidval  28874  grpoinvfval  28884  pjhfval  29758  cvmliftlem5  33251  cvmliftlem15  33260  scutval  33994  trlfset  38174  dicffval  39188  dicfval  39189  dihffval  39244  dihfval  39245  hvmapffval  39772  hvmapfval  39773  hdmap1fval  39810  hdmapffval  39840  hdmapfval  39841  hgmapfval  39900  wessf1ornlem  42722