Theorem riotabidv 5835

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 473	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐴 ∧ 𝜒)))
4	3	iotabidv 5201	. 2 ⊢ (𝜑 → (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜒)))
5		df-riota 5833	. 2 ⊢ (℩𝑥 ∈ 𝐴 𝜓) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))
6		df-riota 5833	. 2 ⊢ (℩𝑥 ∈ 𝐴 𝜒) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜒))
7	4, 5, 6	3eqtr4g 2235	1 ⊢ (𝜑 → (℩𝑥 ∈ 𝐴 𝜓) = (℩𝑥 ∈ 𝐴 𝜒))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1353   ∈ wcel 2148  ℩cio 5178  ℩crio 5832

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-uni 3812  df-iota 5180  df-riota 5833

This theorem is referenced by:  riotaeqbidv  5836  csbriotag  5845  infvalti  7023  caucvgsrlemfv  7792  axcaucvglemval  7898  axcaucvglemcau  7899  subval  8151  divvalap  8633  divfnzn  9623  flval  10274  cjval  10856  sqrtrval  11011  qnumval  12187  qdenval  12188  grpinvval  12921