Theorem fvsb 43201

Description: Explicit substitution of a value of a function into a wff. (Contributed by Andrew Salmon, 1-Aug-2011.)

Assertion

Ref	Expression
fvsb	⊢ (∃!𝑦 𝐴𝐹𝑦 → ([(𝐹‘𝐴) / 𝑥]𝜑 ↔ ∃𝑥(∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑥) ∧ 𝜑)))

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐹,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem fvsb

Step	Hyp	Ref	Expression
1		df-fv 6551	. . 3 ⊢ (𝐹‘𝐴) = (℩𝑦𝐴𝐹𝑦)
2		dfsbcq 3779	. . 3 ⊢ ((𝐹‘𝐴) = (℩𝑦𝐴𝐹𝑦) → ([(𝐹‘𝐴) / 𝑥]𝜑 ↔ [(℩𝑦𝐴𝐹𝑦) / 𝑥]𝜑))
3	1, 2	ax-mp 5	. 2 ⊢ ([(𝐹‘𝐴) / 𝑥]𝜑 ↔ [(℩𝑦𝐴𝐹𝑦) / 𝑥]𝜑)
4		iotasbc 43168	. 2 ⊢ (∃!𝑦 𝐴𝐹𝑦 → ([(℩𝑦𝐴𝐹𝑦) / 𝑥]𝜑 ↔ ∃𝑥(∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑥) ∧ 𝜑)))
5	3, 4	bitrid 282	1 ⊢ (∃!𝑦 𝐴𝐹𝑦 → ([(𝐹‘𝐴) / 𝑥]𝜑 ↔ ∃𝑥(∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑥) ∧ 𝜑)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396  ∀wal 1539   = wceq 1541  ∃wex 1781  ∃!weu 2562  [wsbc 3777   class class class wbr 5148  ℩cio 6493  ‘cfv 6543

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-v 3476  df-sbc 3778  df-un 3953  df-in 3955  df-ss 3965  df-sn 4629  df-pr 4631  df-uni 4909  df-iota 6495  df-fv 6551

This theorem is referenced by:  fveqsb  43202