Theorem sbaniota 45009

Description: Theorem *14.26 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 12-Jul-2011.)

Assertion

Ref	Expression
sbaniota	⊢ (∃!𝑥𝜑 → (∃𝑥(𝜑 ∧ 𝜓) ↔ [(℩𝑥𝜑) / 𝑥]𝜓))

Proof of Theorem sbaniota

Step	Hyp	Ref	Expression
1		eupickbi 2666	. 2 ⊢ (∃!𝑥𝜑 → (∃𝑥(𝜑 ∧ 𝜓) ↔ ∀𝑥(𝜑 → 𝜓)))
2		sbiota1 45008	. 2 ⊢ (∃!𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) ↔ [(℩𝑥𝜑) / 𝑥]𝜓))
3	1, 2	bitrd 282	1 ⊢ (∃!𝑥𝜑 → (∃𝑥(𝜑 ∧ 𝜓) ↔ [(℩𝑥𝜑) / 𝑥]𝜓))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1561  ∃wex 1802  ∃!weu 2598  [wsbc 3747  ℩cio 6479

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1566  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ral 3080  df-rex 3090  df-v 3459  df-sbc 3748  df-un 3912  df-ss 3924  df-sn 4586  df-pr 4588  df-uni 4869  df-iota 6481

This theorem is referenced by: (None)