Metamath Proof Explorer


Theorem csbeq2dv

Description: Formula-building deduction for class substitution. (Contributed by NM, 10-Nov-2005) (Revised by Mario Carneiro, 1-Sep-2015)

Ref Expression
Hypothesis csbeq2dv.1 φ B = C
Assertion csbeq2dv φ A / x B = A / x C

Proof

Step Hyp Ref Expression
1 csbeq2dv.1 φ B = C
2 1 eleq2d φ y B y C
3 2 sbcbidv φ [˙A / x]˙ y B [˙A / x]˙ y C
4 3 abbidv φ y | [˙A / x]˙ y B = y | [˙A / x]˙ y C
5 df-csb A / x B = y | [˙A / x]˙ y B
6 df-csb A / x C = y | [˙A / x]˙ y C
7 4 5 6 3eqtr4g φ A / x B = A / x C