Theorem pm14.18 39588

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 39578	. 2 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ V)
2		spsbc 3665	. 2 ⊢ ((℩𝑥𝜑) ∈ V → (∀𝑥𝜓 → [(℩𝑥𝜑) / 𝑥]𝜓))
3	1, 2	syl 17	1 ⊢ (∃!𝑥𝜑 → (∀𝑥𝜓 → [(℩𝑥𝜑) / 𝑥]𝜓))

Syntax hints:   → wi 4  ∀wal 1599   ∈ wcel 2107  ∃!weu 2586  Vcvv 3398  [wsbc 3652  ℩cio 6097

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-rex 3096  df-v 3400  df-sbc 3653  df-un 3797  df-sn 4399  df-pr 4401  df-uni 4672  df-iota 6099

This theorem is referenced by: (None)