Theorem iotasbc5 45182

Description: Theorem *14.205 in [WhiteheadRussell] p. 190. (Contributed by Andrew Salmon, 11-Jul-2011.)

Assertion

Ref	Expression
iotasbc5	⊢ (∃!𝑥𝜑 → ([(℩𝑥𝜑) / 𝑦]𝜓 ↔ ∃𝑦(𝑦 = (℩𝑥𝜑) ∧ 𝜓)))

Distinct variable groups:   𝑥,𝑦   𝜑,𝑦

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥, 𝑦)

Proof of Theorem iotasbc5

Step	Hyp	Ref	Expression
1		sbc5 3775	. 2 ⊢ ([(℩𝑥𝜑) / 𝑦]𝜓 ↔ ∃𝑦(𝑦 = (℩𝑥𝜑) ∧ 𝜓))
2	1	a1i 11	1 ⊢ (∃!𝑥𝜑 → ([(℩𝑥𝜑) / 𝑦]𝜓 ↔ ∃𝑦(𝑦 = (℩𝑥𝜑) ∧ 𝜓)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812  ∃!weu 2599  [wsbc 3747  ℩cio 6497

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-12 2216  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-sbc 3748

This theorem is used by: (None)