This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem nfreud

Description: Deduction version of nfreu . Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 15-Feb-2013) (Revised by Mario Carneiro, 8-Oct-2016) (New usage is discouraged.)

Ref Expression
Hypotheses nfrmod.1 ⊢ Ⅎ 𝑦 𝜑
nfrmod.2 ⊢ ( 𝜑 → Ⅎ 𝑥 𝐴 )
nfrmod.3 ⊢ ( 𝜑 → Ⅎ 𝑥 𝜓 )
Assertion nfreud ( 𝜑 → Ⅎ 𝑥 ∃! 𝑦 ∈ 𝐴 𝜓 )

Proof

Step Hyp Ref Expression
1 nfrmod.1 ⊢ Ⅎ 𝑦 𝜑
2 nfrmod.2 ⊢ ( 𝜑 → Ⅎ 𝑥 𝐴 )
3 nfrmod.3 ⊢ ( 𝜑 → Ⅎ 𝑥 𝜓 )
4 df-reu ⊢ ( ∃! 𝑦 ∈ 𝐴 𝜓 ↔ ∃! 𝑦 ( 𝑦 ∈ 𝐴 ∧ 𝜓 ) )
5 nfcvf ⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → Ⅎ 𝑥 𝑦 )
6 5 adantl ⊢ ( ( 𝜑 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ) → Ⅎ 𝑥 𝑦 )
7 2 adantr ⊢ ( ( 𝜑 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ) → Ⅎ 𝑥 𝐴 )
8 6 7 nfeld ⊢ ( ( 𝜑 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ) → Ⅎ 𝑥 𝑦 ∈ 𝐴 )
9 3 adantr ⊢ ( ( 𝜑 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ) → Ⅎ 𝑥 𝜓 )
10 8 9 nfand ⊢ ( ( 𝜑 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ) → Ⅎ 𝑥 ( 𝑦 ∈ 𝐴 ∧ 𝜓 ) )
11 1 10 nfeud2 ⊢ ( 𝜑 → Ⅎ 𝑥 ∃! 𝑦 ( 𝑦 ∈ 𝐴 ∧ 𝜓 ) )
12 4 11 nfxfrd ⊢ ( 𝜑 → Ⅎ 𝑥 ∃! 𝑦 ∈ 𝐴 𝜓 )