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Theorem iota2 6038
Description: The unique element such that 𝜑. (Contributed by Jeff Madsen, 1-Jun-2011.) (Revised by Mario Carneiro, 23-Dec-2016.)
Hypothesis
Ref Expression
iota2.1 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
iota2 ((𝐴𝐵 ∧ ∃!𝑥𝜑) → (𝜓 ↔ (℩𝑥𝜑) = 𝐴))
Distinct variable groups:   𝑥,𝐴   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)

Proof of Theorem iota2
StepHypRef Expression
1 elex 3352 . 2 (𝐴𝐵𝐴 ∈ V)
2 simpl 474 . . 3 ((𝐴 ∈ V ∧ ∃!𝑥𝜑) → 𝐴 ∈ V)
3 simpr 479 . . 3 ((𝐴 ∈ V ∧ ∃!𝑥𝜑) → ∃!𝑥𝜑)
4 iota2.1 . . . 4 (𝑥 = 𝐴 → (𝜑𝜓))
54adantl 473 . . 3 (((𝐴 ∈ V ∧ ∃!𝑥𝜑) ∧ 𝑥 = 𝐴) → (𝜑𝜓))
6 nfv 1992 . . . 4 𝑥 𝐴 ∈ V
7 nfeu1 2617 . . . 4 𝑥∃!𝑥𝜑
86, 7nfan 1977 . . 3 𝑥(𝐴 ∈ V ∧ ∃!𝑥𝜑)
9 nfvd 1993 . . 3 ((𝐴 ∈ V ∧ ∃!𝑥𝜑) → Ⅎ𝑥𝜓)
10 nfcvd 2903 . . 3 ((𝐴 ∈ V ∧ ∃!𝑥𝜑) → 𝑥𝐴)
112, 3, 5, 8, 9, 10iota2df 6036 . 2 ((𝐴 ∈ V ∧ ∃!𝑥𝜑) → (𝜓 ↔ (℩𝑥𝜑) = 𝐴))
121, 11sylan 489 1 ((𝐴𝐵 ∧ ∃!𝑥𝜑) → (𝜓 ↔ (℩𝑥𝜑) = 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  ∃!weu 2607  Vcvv 3340  cio 6010
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-v 3342  df-sbc 3577  df-un 3720  df-sn 4322  df-pr 4324  df-uni 4589  df-iota 6012
This theorem is referenced by:  pczpre  15754  pcdiv  15759  rngurd  30097  nosupno  32155  nosupfv  32158  bj-nuliota  33325  unirep  33820  ellimciota  40349
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