Theorem iota4an 6474

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 6473	. 2 ⊢ (∃!𝑥(𝜑 ∧ 𝜓) → [(℩𝑥(𝜑 ∧ 𝜓)) / 𝑥](𝜑 ∧ 𝜓))
2		iotaex 6468	. . . 4 ⊢ (℩𝑥(𝜑 ∧ 𝜓)) ∈ V
3		simpl 483	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝜑)
4	3	sbcth 3745	. . . 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 3778	. . . 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 231	. 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 207   ∧ wa 396   ∈ wcel 2119  ∃!weu 2572  Vcvv 3432  [wsbc 3730  ℩cio 6446

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-12 2189  ax-ext 2712  ax-nul 5235

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-ne 2936  df-v 3434  df-sbc 3731  df-dif 3893  df-un 3895  df-ss 3907  df-nul 4269  df-sn 4563  df-pr 4565  df-uni 4846  df-iota 6448

This theorem is referenced by: (None)