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Metamath Proof Explorer

Theorem sbco2d

Description: A composition law for substitution. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 2-Jun-1993) (Revised by Mario Carneiro, 6-Oct-2016) (New usage is discouraged.)

		Ref	Expression
	Hypotheses	sbco2d.1	⊢ Ⅎ 𝑥 𝜑
		sbco2d.2	⊢ Ⅎ 𝑧 𝜑
		sbco2d.3	⊢ ( 𝜑 → Ⅎ 𝑧 𝜓 )
	Assertion	sbco2d	⊢ ( 𝜑 → ( [ 𝑦 / 𝑧 ] [ 𝑧 / 𝑥 ] 𝜓 ↔ [ 𝑦 / 𝑥 ] 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		sbco2d.1	⊢ Ⅎ 𝑥 𝜑
2		sbco2d.2	⊢ Ⅎ 𝑧 𝜑
3		sbco2d.3	⊢ ( 𝜑 → Ⅎ 𝑧 𝜓 )
4	2 3	nfim1	⊢ Ⅎ 𝑧 ( 𝜑 → 𝜓 )
5	4	sbco2	⊢ ( [ 𝑦 / 𝑧 ] [ 𝑧 / 𝑥 ] ( 𝜑 → 𝜓 ) ↔ [ 𝑦 / 𝑥 ] ( 𝜑 → 𝜓 ) )
6	1	sbrim	⊢ ( [ 𝑧 / 𝑥 ] ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → [ 𝑧 / 𝑥 ] 𝜓 ) )
7	6	sbbii	⊢ ( [ 𝑦 / 𝑧 ] [ 𝑧 / 𝑥 ] ( 𝜑 → 𝜓 ) ↔ [ 𝑦 / 𝑧 ] ( 𝜑 → [ 𝑧 / 𝑥 ] 𝜓 ) )
8	2	sbrim	⊢ ( [ 𝑦 / 𝑧 ] ( 𝜑 → [ 𝑧 / 𝑥 ] 𝜓 ) ↔ ( 𝜑 → [ 𝑦 / 𝑧 ] [ 𝑧 / 𝑥 ] 𝜓 ) )
9	7 8	bitri	⊢ ( [ 𝑦 / 𝑧 ] [ 𝑧 / 𝑥 ] ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → [ 𝑦 / 𝑧 ] [ 𝑧 / 𝑥 ] 𝜓 ) )
10	1	sbrim	⊢ ( [ 𝑦 / 𝑥 ] ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → [ 𝑦 / 𝑥 ] 𝜓 ) )
11	5 9 10	3bitr3i	⊢ ( ( 𝜑 → [ 𝑦 / 𝑧 ] [ 𝑧 / 𝑥 ] 𝜓 ) ↔ ( 𝜑 → [ 𝑦 / 𝑥 ] 𝜓 ) )
12	11	pm5.74ri	⊢ ( 𝜑 → ( [ 𝑦 / 𝑧 ] [ 𝑧 / 𝑥 ] 𝜓 ↔ [ 𝑦 / 𝑥 ] 𝜓 ) )