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Metamath Proof Explorer

Theorem injust

Description: Soundness justification theorem for df-in . (Contributed by Rodolfo Medina, 28-Apr-2010) (Proof shortened by Andrew Salmon, 9-Jul-2011)

		Ref	Expression
	Assertion	injust	⊢ { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵 ) } = { 𝑦 ∣ ( 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) }

Proof

Step	Hyp	Ref	Expression
1		eleq1w	⊢ ( 𝑥 = 𝑧 → ( 𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴 ) )
2		eleq1w	⊢ ( 𝑥 = 𝑧 → ( 𝑥 ∈ 𝐵 ↔ 𝑧 ∈ 𝐵 ) )
3	1 2	anbi12d	⊢ ( 𝑥 = 𝑧 → ( ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵 ) ↔ ( 𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵 ) ) )
4	3	cbvabv	⊢ { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵 ) } = { 𝑧 ∣ ( 𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵 ) }
5		eleq1w	⊢ ( 𝑧 = 𝑦 → ( 𝑧 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴 ) )
6		eleq1w	⊢ ( 𝑧 = 𝑦 → ( 𝑧 ∈ 𝐵 ↔ 𝑦 ∈ 𝐵 ) )
7	5 6	anbi12d	⊢ ( 𝑧 = 𝑦 → ( ( 𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵 ) ↔ ( 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ) )
8	7	cbvabv	⊢ { 𝑧 ∣ ( 𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵 ) } = { 𝑦 ∣ ( 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) }
9	4 8	eqtri	⊢ { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵 ) } = { 𝑦 ∣ ( 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) }