Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  riotabidva Structured version   Visualization version   GIF version

Theorem riotabidva 7110
 Description: Equivalent wff's yield equal restricted class abstractions (deduction form). (rabbidva 3457 analog.) (Contributed by NM, 17-Jan-2012.)
Hypothesis
Ref Expression
riotabidva.1 ((𝜑𝑥𝐴) → (𝜓𝜒))
Assertion
Ref Expression
riotabidva (𝜑 → (𝑥𝐴 𝜓) = (𝑥𝐴 𝜒))
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)   𝐴(𝑥)

Proof of Theorem riotabidva
StepHypRef Expression
1 riotabidva.1 . . . 4 ((𝜑𝑥𝐴) → (𝜓𝜒))
21pm5.32da 581 . . 3 (𝜑 → ((𝑥𝐴𝜓) ↔ (𝑥𝐴𝜒)))
32iotabidv 6315 . 2 (𝜑 → (℩𝑥(𝑥𝐴𝜓)) = (℩𝑥(𝑥𝐴𝜒)))
4 df-riota 7091 . 2 (𝑥𝐴 𝜓) = (℩𝑥(𝑥𝐴𝜓))
5 df-riota 7091 . 2 (𝑥𝐴 𝜒) = (℩𝑥(𝑥𝐴𝜒))
63, 4, 53eqtr4g 2880 1 (𝜑 → (𝑥𝐴 𝜓) = (𝑥𝐴 𝜒))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ℩cio 6288  ℩crio 7090 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-v 3475  df-in 3920  df-ss 3930  df-uni 4815  df-iota 6290  df-riota 7091 This theorem is referenced by:  riotabiia  7111  dfceil2  13193  cidpropd  16959  grpinvpropd  18153  mirval  26428  mirfv  26429  grpoidval  28275  adjval2  29653  xdivval  30582  toslub  30642  tosglb  30644  ringinvval  30871  glbconN  36549  cdlemk33N  38081  cdlemk34  38082  cdlemkid4  38106
 Copyright terms: Public domain W3C validator