This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem nfcsb

Description: Bound-variable hypothesis builder for substitution into a class. Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker nfcsbw when possible. (Contributed by Mario Carneiro, 12-Oct-2016) (New usage is discouraged.)

	Ref	Expression
Hypotheses	nfcsb.1	⊢ Ⅎ 𝑥 𝐴
	nfcsb.2	⊢ Ⅎ 𝑥 𝐵
Assertion	nfcsb	⊢ Ⅎ 𝑥 ⦋ 𝐴 / 𝑦 ⦌ 𝐵

Proof

Step	Hyp	Ref	Expression
1		nfcsb.1	⊢ Ⅎ 𝑥 𝐴
2		nfcsb.2	⊢ Ⅎ 𝑥 𝐵
3		nftru	⊢ Ⅎ 𝑦 ⊤
4	1	a1i	⊢ ( ⊤ → Ⅎ 𝑥 𝐴 )
5	2	a1i	⊢ ( ⊤ → Ⅎ 𝑥 𝐵 )
6	3 4 5	nfcsbd	⊢ ( ⊤ → Ⅎ 𝑥 ⦋ 𝐴 / 𝑦 ⦌ 𝐵 )
7	6	mptru	⊢ Ⅎ 𝑥 ⦋ 𝐴 / 𝑦 ⦌ 𝐵