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Metamath Proof Explorer

Theorem addsubeq4d

Description: Relation between sums and differences. (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 addsubeq4d ( 𝜑 → ( ( 𝐴 + 𝐵 ) = ( 𝐶 + 𝐷 ) ↔ ( 𝐶 − 𝐴 ) = ( 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 addsubeq4 ⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ ( 𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ ) ) → ( ( 𝐴 + 𝐵 ) = ( 𝐶 + 𝐷 ) ↔ ( 𝐶 − 𝐴 ) = ( 𝐵 − 𝐷 ) ) )
6 1 2 3 4 5 syl22anc ⊢ ( 𝜑 → ( ( 𝐴 + 𝐵 ) = ( 𝐶 + 𝐷 ) ↔ ( 𝐶 − 𝐴 ) = ( 𝐵 − 𝐷 ) ) )