Theorem iota4an 6520

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 6519	. 2 ⊢ (∃!𝑥(𝜑 ∧ 𝜓) → [(℩𝑥(𝜑 ∧ 𝜓)) / 𝑥](𝜑 ∧ 𝜓))
2		iotaex 6514	. . . 4 ⊢ (℩𝑥(𝜑 ∧ 𝜓)) ∈ V
3		simpl 487	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝜑)
4	3	sbcth 3760	. . . 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 3793	. . . 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 233	. 2 ⊢ ([(℩𝑥(𝜑 ∧ 𝜓)) / 𝑥](𝜑 ∧ 𝜓) → [(℩𝑥(𝜑 ∧ 𝜓)) / 𝑥]𝜑)
9	1, 8	syl 18	1 ⊢ (∃!𝑥(𝜑 ∧ 𝜓) → [(℩𝑥(𝜑 ∧ 𝜓)) / 𝑥]𝜑)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∈ wcel 2143  ∃!weu 2596  Vcvv 3455  [wsbc 3745  ℩cio 6492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-12 2213  ax-ext 2735  ax-nul 5270

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-v 3457  df-sbc 3746  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4288  df-sn 4591  df-pr 4593  df-uni 4874  df-iota 6494

This theorem is referenced by: (None)