Theorem fveqsb 45109

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 6894	. . 3 ⊢ (𝐹‘𝐴) ∈ V
2		fveqsb.3	. . . 4 ⊢ Ⅎ𝑥𝜓
3		fveqsb.2	. . . 4 ⊢ (𝑥 = (𝐹‘𝐴) → (𝜑 ↔ 𝜓))
4	2, 3	sbciegf 3781	. . 3 ⊢ ((𝐹‘𝐴) ∈ V → ([(𝐹‘𝐴) / 𝑥]𝜑 ↔ 𝜓))
5	1, 4	ax-mp 5	. 2 ⊢ ([(𝐹‘𝐴) / 𝑥]𝜑 ↔ 𝜓)
6		fvsb 45108	. 2 ⊢ (∃!𝑦 𝐴𝐹𝑦 → ([(𝐹‘𝐴) / 𝑥]𝜑 ↔ ∃𝑥(∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑥) ∧ 𝜑)))
7	5, 6	bitr3id 288	1 ⊢ (∃!𝑦 𝐴𝐹𝑦 → (𝜓 ↔ ∃𝑥(∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑥) ∧ 𝜑)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1566   = wceq 1568  ∃wex 1807  Ⅎwnf 1811   ∈ wcel 2141  ∃!weu 2594  Vcvv 3453  [wsbc 3743   class class class wbr 5108  ‘cfv 6536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-12 2211  ax-ext 2733  ax-nul 5268

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-v 3455  df-sbc 3744  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-sn 4589  df-pr 4591  df-uni 4872  df-iota 6492  df-fv 6544

This theorem is referenced by: (None)