Theorem fvsb 41959

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 6426	. . 3 ⊢ (𝐹‘𝐴) = (℩𝑦𝐴𝐹𝑦)
2		dfsbcq 3713	. . 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 41926	. 2 ⊢ (∃!𝑦 𝐴𝐹𝑦 → ([(℩𝑦𝐴𝐹𝑦) / 𝑥]𝜑 ↔ ∃𝑥(∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑥) ∧ 𝜑)))
5	3, 4	syl5bb 282	1 ⊢ (∃!𝑦 𝐴𝐹𝑦 → ([(𝐹‘𝐴) / 𝑥]𝜑 ↔ ∃𝑥(∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑥) ∧ 𝜑)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395  ∀wal 1537   = wceq 1539  ∃wex 1783  ∃!weu 2568  [wsbc 3711   class class class wbr 5070  ℩cio 6374  ‘cfv 6418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-v 3424  df-sbc 3712  df-un 3888  df-in 3890  df-ss 3900  df-sn 4559  df-pr 4561  df-uni 4837  df-iota 6376  df-fv 6426

This theorem is referenced by:  fveqsb  41960