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Theorem riotabidv 5413
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 442 . . 3 (φ → ((x A ψ) ↔ (x A χ)))
43iotabidv 4831 . 2 (φ → (℩x(x A ψ)) = (℩x(x A χ)))
5 df-riota 5411 . 2 (x A ψ) = (℩x(x A ψ))
6 df-riota 5411 . 2 (x A χ) = (℩x(x A χ))
74, 5, 63eqtr4g 2094 1 (φ → (x A ψ) = (x A χ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242   wcel 1390  cio 4808  crio 5410
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-uni 3572  df-iota 4810  df-riota 5411
This theorem is referenced by:  riotaeqbidv  5414  csbriotag  5423  subval  6960  divvalap  7395  divfnzn  8292  cjval  9033
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