Theorem iota4an 6497

Description: Theorem *14.23 in [WhiteheadRussell] p. 191. (Contributed by Andrew Salmon, 12-Jul-2011.)

Assertion

Ref	Expression
iota4an	⊢ (∃!𝑥(𝜑 ∧ 𝜓) → [(℩𝑥(𝜑 ∧ 𝜓)) / 𝑥]𝜑)

Proof of Theorem iota4an

Step	Hyp	Ref	Expression
1		iota4 6496	. 2 ⊢ (∃!𝑥(𝜑 ∧ 𝜓) → [(℩𝑥(𝜑 ∧ 𝜓)) / 𝑥](𝜑 ∧ 𝜓))
2		iotaex 6491	. . . 4 ⊢ (℩𝑥(𝜑 ∧ 𝜓)) ∈ V
3		simpl 486	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝜑)
4	3	sbcth 3757	. . . 4 ⊢ ((℩𝑥(𝜑 ∧ 𝜓)) ∈ V → [(℩𝑥(𝜑 ∧ 𝜓)) / 𝑥]((𝜑 ∧ 𝜓) → 𝜑))
5	2, 4	ax-mp 5	. . 3 ⊢ [(℩𝑥(𝜑 ∧ 𝜓)) / 𝑥]((𝜑 ∧ 𝜓) → 𝜑)
6		sbcimg 3790	. . . 4 ⊢ ((℩𝑥(𝜑 ∧ 𝜓)) ∈ V → ([(℩𝑥(𝜑 ∧ 𝜓)) / 𝑥]((𝜑 ∧ 𝜓) → 𝜑) ↔ ([(℩𝑥(𝜑 ∧ 𝜓)) / 𝑥](𝜑 ∧ 𝜓) → [(℩𝑥(𝜑 ∧ 𝜓)) / 𝑥]𝜑)))
7	2, 6	ax-mp 5	. . 3 ⊢ ([(℩𝑥(𝜑 ∧ 𝜓)) / 𝑥]((𝜑 ∧ 𝜓) → 𝜑) ↔ ([(℩𝑥(𝜑 ∧ 𝜓)) / 𝑥](𝜑 ∧ 𝜓) → [(℩𝑥(𝜑 ∧ 𝜓)) / 𝑥]𝜑))
8	5, 7	mpbi 232	. 2 ⊢ ([(℩𝑥(𝜑 ∧ 𝜓)) / 𝑥](𝜑 ∧ 𝜓) → [(℩𝑥(𝜑 ∧ 𝜓)) / 𝑥]𝜑)
9	1, 8	syl 17	1 ⊢ (∃!𝑥(𝜑 ∧ 𝜓) → [(℩𝑥(𝜑 ∧ 𝜓)) / 𝑥]𝜑)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∈ wcel 2141  ∃!weu 2594  Vcvv 3453  [wsbc 3742  ℩cio 6469

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-12 2211  ax-ext 2733  ax-nul 5253

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-v 3455  df-sbc 3743  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4284  df-sn 4580  df-pr 4582  df-uni 4863  df-iota 6471

This theorem is referenced by: (None)