Theorem pm14.18 44995

Description: Theorem *14.18 in [WhiteheadRussell] p. 189. (Contributed by Andrew Salmon, 11-Jul-2011.)

Assertion

Ref	Expression
pm14.18	⊢ (∃!𝑥𝜑 → (∀𝑥𝜓 → [(℩𝑥𝜑) / 𝑥]𝜓))

Proof of Theorem pm14.18

Step	Hyp	Ref	Expression
1		iotaexeu 44985	. 2 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ V)
2		spsbc 3758	. 2 ⊢ ((℩𝑥𝜑) ∈ V → (∀𝑥𝜓 → [(℩𝑥𝜑) / 𝑥]𝜓))
3	1, 2	syl 17	1 ⊢ (∃!𝑥𝜑 → (∀𝑥𝜓 → [(℩𝑥𝜑) / 𝑥]𝜓))

Syntax hints:   → wi 4  ∀wal 1559   ∈ wcel 2143  ∃!weu 2596  Vcvv 3455  [wsbc 3745  ℩cio 6475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-12 2213  ax-ext 2735

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1564  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-v 3457  df-sbc 3746  df-un 3910  df-ss 3922  df-sn 4584  df-pr 4586  df-uni 4867  df-iota 6477

This theorem is referenced by: (None)