Theorem csbeq2i 3174

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

Hypothesis

Ref	Expression
csbeq2i.1	⊢ 𝐵 = 𝐶

Assertion

Ref	Expression
csbeq2i	⊢ ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶

Proof of Theorem csbeq2i

Step	Hyp	Ref	Expression
1		csbeq2i.1	. . . 4 ⊢ 𝐵 = 𝐶
2	1	a1i 9	. . 3 ⊢ (⊤ → 𝐵 = 𝐶)
3	2	csbeq2dv 3173	. 2 ⊢ (⊤ → ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶)
4	3	mptru 1411	1 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶

Syntax hints:   = wceq 1402  ⊤wtru 1403  ⦋csb 3147

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-sbc 3052  df-csb 3148

This theorem is referenced by:  csbvarg  3175  csbnest1g  3203  csbsng  3766  csbunig  3938  csbxpg  4851  csbcnvg  4959  csbdmg  4970  csbresg  5061  csbrng  5244  csbfv12g  5730  csbnegg  8514  iseqf1olemjpcl  10923  iseqf1olemqpcl  10924  iseqf1olemfvp  10925  seq3f1olemqsum  10928  csbwrdg  11312