Theorem iota4an 6470

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 6469	. 2 ⊢ (∃!𝑥(𝜑 ∧ 𝜓) → [(℩𝑥(𝜑 ∧ 𝜓)) / 𝑥](𝜑 ∧ 𝜓))
2		iotaex 6464	. . . 4 ⊢ (℩𝑥(𝜑 ∧ 𝜓)) ∈ V
3		simpl 482	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝜑)
4	3	sbcth 3752	. . . 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 3786	. . . 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 230	. 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 206   ∧ wa 395   ∈ wcel 2113  ∃!weu 2565  Vcvv 3437  [wsbc 3737  ℩cio 6442

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-12 2182  ax-ext 2705  ax-nul 5248

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-ne 2930  df-v 3439  df-sbc 3738  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-sn 4578  df-pr 4580  df-uni 4861  df-iota 6444

This theorem is referenced by: (None)