Theorem pm14.18 44611

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 44601	. 2 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ V)
2		spsbc 3751	. 2 ⊢ ((℩𝑥𝜑) ∈ V → (∀𝑥𝜓 → [(℩𝑥𝜑) / 𝑥]𝜓))
3	1, 2	syl 17	1 ⊢ (∃!𝑥𝜑 → (∀𝑥𝜓 → [(℩𝑥𝜑) / 𝑥]𝜓))

Syntax hints:   → wi 4  ∀wal 1539   ∈ wcel 2113  ∃!weu 2566  Vcvv 3438  [wsbc 3738  ℩cio 6444

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 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-v 3440  df-sbc 3739  df-un 3904  df-ss 3916  df-sn 4579  df-pr 4581  df-uni 4862  df-iota 6446

This theorem is referenced by: (None)