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Theorem riotabidv 5395
Description: Formula-building deduction rule for restricted iota. (Contributed by NM, 15-Sep-2011.)
Hypothesis
Ref Expression
riotabidv.1 (φ → (ψχ))
Assertion
Ref Expression
riotabidv (φ → (x A ψ) = (x A χ))
Distinct variable group:   φ,x
Allowed substitution hints:   ψ(x)   χ(x)   A(x)

Proof of Theorem riotabidv
StepHypRef Expression
1 biidd 161 . . . 4 (φ → (x Ax A))
2 riotabidv.1 . . . 4 (φ → (ψχ))
31, 2anbi12d 445 . . 3 (φ → ((x A ψ) ↔ (x A χ)))
43iotabidv 4815 . 2 (φ → (℩x(x A ψ)) = (℩x(x A χ)))
5 df-riota 5393 . 2 (x A ψ) = (℩x(x A ψ))
6 df-riota 5393 . 2 (x A χ) = (℩x(x A χ))
74, 5, 63eqtr4g 2079 1 (φ → (x A ψ) = (x A χ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1228   wcel 1374  cio 4792  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-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-rex 2290  df-uni 3555  df-iota 4794  df-riota 5393
This theorem is referenced by:  riotaeqbidv  5396  csbriotag  5404  subval  6796
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