Theorem fvsb 44492

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 6489	. . 3 ⊢ (𝐹‘𝐴) = (℩𝑦𝐴𝐹𝑦)
2		dfsbcq 3738	. . 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 44460	. 2 ⊢ (∃!𝑦 𝐴𝐹𝑦 → ([(℩𝑦𝐴𝐹𝑦) / 𝑥]𝜑 ↔ ∃𝑥(∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑥) ∧ 𝜑)))
5	3, 4	bitrid 283	1 ⊢ (∃!𝑦 𝐴𝐹𝑦 → ([(𝐹‘𝐴) / 𝑥]𝜑 ↔ ∃𝑥(∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑥) ∧ 𝜑)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1539   = wceq 1541  ∃wex 1780  ∃!weu 2563  [wsbc 3736   class class class wbr 5089  ℩cio 6435  ‘cfv 6481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-12 2180  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-v 3438  df-sbc 3737  df-un 3902  df-ss 3914  df-sn 4574  df-pr 4576  df-uni 4857  df-iota 6437  df-fv 6489

This theorem is referenced by:  fveqsb  44493