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Metamath Proof Explorer

Theorem sub4d

Description: Rearrangement of 4 terms in a subtraction. (Contributed by Mario Carneiro, 27-May-2016)

Ref Expression
Hypotheses negidd.1 ⊢ ( 𝜑 → 𝐴 ∈ ℂ )
pncand.2 ⊢ ( 𝜑 → 𝐵 ∈ ℂ )
subaddd.3 ⊢ ( 𝜑 → 𝐶 ∈ ℂ )
addsub4d.4 ⊢ ( 𝜑 → 𝐷 ∈ ℂ )
Assertion sub4d ( 𝜑 → ( ( 𝐴 − 𝐵 ) − ( 𝐶 − 𝐷 ) ) = ( ( 𝐴 − 𝐶 ) − ( 𝐵 − 𝐷 ) ) )

Proof

Step Hyp Ref Expression
1 negidd.1 ⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2 pncand.2 ⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3 subaddd.3 ⊢ ( 𝜑 → 𝐶 ∈ ℂ )
4 addsub4d.4 ⊢ ( 𝜑 → 𝐷 ∈ ℂ )
5 sub4 ⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ ( 𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ ) ) → ( ( 𝐴 − 𝐵 ) − ( 𝐶 − 𝐷 ) ) = ( ( 𝐴 − 𝐶 ) − ( 𝐵 − 𝐷 ) ) )
6 1 2 3 4 5 syl22anc ⊢ ( 𝜑 → ( ( 𝐴 − 𝐵 ) − ( 𝐶 − 𝐷 ) ) = ( ( 𝐴 − 𝐶 ) − ( 𝐵 − 𝐷 ) ) )