Theorem sbaniota 44459

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 2635	. 2 ⊢ (∃!𝑥𝜑 → (∃𝑥(𝜑 ∧ 𝜓) ↔ ∀𝑥(𝜑 → 𝜓)))
2		sbiota1 44458	. 2 ⊢ (∃!𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) ↔ [(℩𝑥𝜑) / 𝑥]𝜓))
3	1, 2	bitrd 279	1 ⊢ (∃!𝑥𝜑 → (∃𝑥(𝜑 ∧ 𝜓) ↔ [(℩𝑥𝜑) / 𝑥]𝜓))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1537  ∃wex 1778  ∃!weu 2567  [wsbc 3787  ℩cio 6511

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ral 3061  df-rex 3070  df-v 3481  df-sbc 3788  df-un 3955  df-ss 3967  df-sn 4626  df-pr 4628  df-uni 4907  df-iota 6513

This theorem is referenced by: (None)