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Metamath Proof Explorer

Theorem nfifd

Description: Deduction form of nfif . (Contributed by NM, 15-Feb-2013) (Revised by Mario Carneiro, 13-Oct-2016)

Ref Expression
Hypotheses nfifd.2 ⊢ ( 𝜑 → Ⅎ 𝑥 𝜓 )
nfifd.3 ⊢ ( 𝜑 → Ⅎ 𝑥 𝐴 )
nfifd.4 ⊢ ( 𝜑 → Ⅎ 𝑥 𝐵 )
Assertion nfifd ( 𝜑 → Ⅎ 𝑥 if ( 𝜓 , 𝐴 , 𝐵 ) )

Proof

Step Hyp Ref Expression
1 nfifd.2 ⊢ ( 𝜑 → Ⅎ 𝑥 𝜓 )
2 nfifd.3 ⊢ ( 𝜑 → Ⅎ 𝑥 𝐴 )
3 nfifd.4 ⊢ ( 𝜑 → Ⅎ 𝑥 𝐵 )
4 dfif2 ⊢ if ( 𝜓 , 𝐴 , 𝐵 ) = { 𝑦 ∣ ( ( 𝑦 ∈ 𝐵 → 𝜓 ) → ( 𝑦 ∈ 𝐴 ∧ 𝜓 ) ) }
5 nfv ⊢ Ⅎ 𝑦 𝜑
6 3 nfcrd ⊢ ( 𝜑 → Ⅎ 𝑥 𝑦 ∈ 𝐵 )
7 6 1 nfimd ⊢ ( 𝜑 → Ⅎ 𝑥 ( 𝑦 ∈ 𝐵 → 𝜓 ) )
8 2 nfcrd ⊢ ( 𝜑 → Ⅎ 𝑥 𝑦 ∈ 𝐴 )
9 8 1 nfand ⊢ ( 𝜑 → Ⅎ 𝑥 ( 𝑦 ∈ 𝐴 ∧ 𝜓 ) )
10 7 9 nfimd ⊢ ( 𝜑 → Ⅎ 𝑥 ( ( 𝑦 ∈ 𝐵 → 𝜓 ) → ( 𝑦 ∈ 𝐴 ∧ 𝜓 ) ) )
11 5 10 nfabdw ⊢ ( 𝜑 → Ⅎ 𝑥 { 𝑦 ∣ ( ( 𝑦 ∈ 𝐵 → 𝜓 ) → ( 𝑦 ∈ 𝐴 ∧ 𝜓 ) ) } )
12 4 11 nfcxfrd ⊢ ( 𝜑 → Ⅎ 𝑥 if ( 𝜓 , 𝐴 , 𝐵 ) )