Theorem pm14.18 38629

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 38619	. 2 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ V)
2		spsbc 3448	. 2 ⊢ ((℩𝑥𝜑) ∈ V → (∀𝑥𝜓 → [(℩𝑥𝜑) / 𝑥]𝜓))
3	1, 2	syl 17	1 ⊢ (∃!𝑥𝜑 → (∀𝑥𝜓 → [(℩𝑥𝜑) / 𝑥]𝜓))

Syntax hints:   → wi 4  ∀wal 1481   ∈ wcel 1990  ∃!weu 2470  Vcvv 3200  [wsbc 3435  ℩cio 5849

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-v 3202  df-sbc 3436  df-un 3579  df-sn 4178  df-pr 4180  df-uni 4437  df-iota 5851

This theorem is referenced by: (None)