Metamath Proof Explorer


Theorem nfcsbw

Description: Bound-variable hypothesis builder for substitution into a class. Version of nfcsb with a disjoint variable condition, which does not require ax-13 . (Contributed by Mario Carneiro, 12-Oct-2016) (Revised by Gino Giotto, 10-Jan-2024)

Ref Expression
Hypotheses nfcsbw.1 _ x A
nfcsbw.2 _ x B
Assertion nfcsbw _ x A / y B

Proof

Step Hyp Ref Expression
1 nfcsbw.1 _ x A
2 nfcsbw.2 _ x B
3 df-csb A / y B = z | [˙A / y]˙ z B
4 nftru z
5 nftru y
6 1 a1i _ x A
7 2 a1i _ x B
8 7 nfcrd x z B
9 5 6 8 nfsbcdw x [˙A / y]˙ z B
10 4 9 nfabdw _ x z | [˙A / y]˙ z B
11 3 10 nfcxfrd _ x A / y B
12 11 mptru _ x A / y B