Theorem iota2d 4367

Description: A condition that allows us to represent "the unique element such that φ " with a class expression A. (Contributed by NM, 30-Dec-2014.)

Hypotheses

Ref	Expression
iota2df.1	⊢ (φ → B ∈ V)
iota2df.2	⊢ (φ → ∃!xψ)
iota2df.3	⊢ ((φ ∧ x = B) → (ψ ↔ χ))

Assertion

Ref	Expression
iota2d	⊢ (φ → (χ ↔ (℩xψ) = B))

Distinct variable groups:   x,B   φ,x   χ,x

Allowed substitution hints:   ψ(x)   V(x)

Proof of Theorem iota2d

Step	Hyp	Ref	Expression
1		iota2df.1	. 2 ⊢ (φ → B ∈ V)
2		iota2df.2	. 2 ⊢ (φ → ∃!xψ)
3		iota2df.3	. 2 ⊢ ((φ ∧ x = B) → (ψ ↔ χ))
4		nfv 1619	. 2 ⊢ Ⅎxφ
5		nfvd 1620	. 2 ⊢ (φ → Ⅎxχ)
6		nfcvd 2491	. 2 ⊢ (φ → ℲxB)
7	1, 2, 3, 4, 5, 6	iota2df 4366	1 ⊢ (φ → (χ ↔ (℩xψ) = B))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∃!weu 2204  ℩cio 4338

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-rex 2621  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-un 3215  df-sn 3742  df-pr 3743  df-uni 3893  df-iota 4340

This theorem is referenced by: (None)