Theorem fveqsb 40783

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

Hypotheses

Ref	Expression
fveqsb.2	⊢ (𝑥 = (𝐹‘𝐴) → (𝜑 ↔ 𝜓))
fveqsb.3	⊢ Ⅎ𝑥𝜓

Assertion

Ref	Expression
fveqsb	⊢ (∃!𝑦 𝐴𝐹𝑦 → (𝜓 ↔ ∃𝑥(∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑥) ∧ 𝜑)))

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

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

Proof of Theorem fveqsb

Step	Hyp	Ref	Expression
1		fvex 6682	. . 3 ⊢ (𝐹‘𝐴) ∈ V
2		fveqsb.3	. . . 4 ⊢ Ⅎ𝑥𝜓
3		fveqsb.2	. . . 4 ⊢ (𝑥 = (𝐹‘𝐴) → (𝜑 ↔ 𝜓))
4	2, 3	sbciegf 3808	. . 3 ⊢ ((𝐹‘𝐴) ∈ V → ([(𝐹‘𝐴) / 𝑥]𝜑 ↔ 𝜓))
5	1, 4	ax-mp 5	. 2 ⊢ ([(𝐹‘𝐴) / 𝑥]𝜑 ↔ 𝜓)
6		fvsb 40782	. 2 ⊢ (∃!𝑦 𝐴𝐹𝑦 → ([(𝐹‘𝐴) / 𝑥]𝜑 ↔ ∃𝑥(∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑥) ∧ 𝜑)))
7	5, 6	syl5bbr 287	1 ⊢ (∃!𝑦 𝐴𝐹𝑦 → (𝜓 ↔ ∃𝑥(∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑥) ∧ 𝜑)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1531   = wceq 1533  ∃wex 1776  Ⅎwnf 1780   ∈ wcel 2110  ∃!weu 2649  Vcvv 3494  [wsbc 3771   class class class wbr 5065  ‘cfv 6354

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-nul 5209

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-sn 4567  df-pr 4569  df-uni 4838  df-iota 6313  df-fv 6362

This theorem is referenced by: (None)