Theorem pm14.18 44425

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 44415	. 2 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ V)
2		spsbc 3800	. 2 ⊢ ((℩𝑥𝜑) ∈ V → (∀𝑥𝜓 → [(℩𝑥𝜑) / 𝑥]𝜓))
3	1, 2	syl 17	1 ⊢ (∃!𝑥𝜑 → (∀𝑥𝜓 → [(℩𝑥𝜑) / 𝑥]𝜓))

Syntax hints:   → wi 4  ∀wal 1538   ∈ wcel 2108  ∃!weu 2567  Vcvv 3479  [wsbc 3787  ℩cio 6510

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-v 3481  df-sbc 3788  df-un 3955  df-ss 3967  df-sn 4625  df-pr 4627  df-uni 4906  df-iota 6512

This theorem is referenced by: (None)