Theorem fvsb 40804

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 6363	. . 3 ⊢ (𝐹‘𝐴) = (℩𝑦𝐴𝐹𝑦)
2		dfsbcq 3774	. . 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 40771	. 2 ⊢ (∃!𝑦 𝐴𝐹𝑦 → ([(℩𝑦𝐴𝐹𝑦) / 𝑥]𝜑 ↔ ∃𝑥(∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑥) ∧ 𝜑)))
5	3, 4	syl5bb 285	1 ⊢ (∃!𝑦 𝐴𝐹𝑦 → ([(𝐹‘𝐴) / 𝑥]𝜑 ↔ ∃𝑥(∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑥) ∧ 𝜑)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1535   = wceq 1537  ∃wex 1780  ∃!weu 2653  [wsbc 3772   class class class wbr 5066  ℩cio 6312  ‘cfv 6355

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-v 3496  df-sbc 3773  df-un 3941  df-in 3943  df-ss 3952  df-sn 4568  df-pr 4570  df-uni 4839  df-iota 6314  df-fv 6363

This theorem is referenced by:  fveqsb  40805