Theorem riotabidv 6040

Description: Formula-building deduction for restricted iota. (Contributed by NM, 15-Sep-2011.)

Hypothesis

Ref	Expression
riotabidv.1	⊢ (𝜑 → (𝜓 ↔ 𝜒))

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

Step	Hyp	Ref	Expression
1		biidd 172	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
2		riotabidv.1	. . . 4 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
3	1, 2	anbi12d 477	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐴 ∧ 𝜒)))
4	3	iotabidv 5360	. 2 ⊢ (𝜑 → (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜒)))
5		df-riota 6038	. 2 ⊢ (℩𝑥 ∈ 𝐴 𝜓) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))
6		df-riota 6038	. 2 ⊢ (℩𝑥 ∈ 𝐴 𝜒) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜒))
7	4, 5, 6	3eqtr4g 2296	1 ⊢ (𝜑 → (℩𝑥 ∈ 𝐴 𝜓) = (℩𝑥 ∈ 𝐴 𝜒))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1402   ∈ wcel 2209  ℩cio 5335  ℩crio 6037

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-uni 3936  df-iota 5337  df-riota 6038

This theorem is used by:  riotaeqbidv  6041  csbriotag  6052  infvalti  7362  caucvgsrlemfv  8158  axcaucvglemval  8264  axcaucvglemcau  8265  subval  8518  divvalap  9005  divfnzn  10023  flval  10709  cjval  11612  sqrtrval  11768  qnumval  12965  qdenval  12966  grpinvval  13850  uspgredg2v  16474  usgredg2v  16477