Theorem nfcsb 3171

Description: Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.)

Hypotheses

Ref	Expression
nfcsb.1	⊢ ℲxA
nfcsb.2	⊢ ℲxB

Assertion

Ref	Expression
nfcsb	⊢ Ⅎx[A / y]B

Proof of Theorem nfcsb

Step	Hyp	Ref	Expression
1		nftru 1554	. . 3 ⊢ Ⅎy ⊤
2		nfcsb.1	. . . 4 ⊢ ℲxA
3	2	a1i 10	. . 3 ⊢ ( ⊤ → ℲxA)
4		nfcsb.2	. . . 4 ⊢ ℲxB
5	4	a1i 10	. . 3 ⊢ ( ⊤ → ℲxB)
6	1, 3, 5	nfcsbd 3170	. 2 ⊢ ( ⊤ → Ⅎx[A / y]B)
7	6	trud 1323	1 ⊢ Ⅎx[A / y]B

Syntax hints:   ⊤ wtru 1316  Ⅎwnfc 2477  [csb 3137

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-sbc 3048  df-csb 3138

This theorem is referenced by:  cbvralcsf  3199  cbvreucsf  3201  cbvrabcsf  3202  fmpt2x  5731