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| Description: Bound-variable hypothesis builder for substitution into a class. Usage of this theorem is discouraged because it depends on ax-13 2376. Use the weaker nfcsbw 3924 when possible. (Contributed by Mario Carneiro, 12-Oct-2016.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| nfcsb.1 | ⊢ Ⅎ𝑥𝐴 | 
| nfcsb.2 | ⊢ Ⅎ𝑥𝐵 | 
| Ref | Expression | 
|---|---|
| nfcsb | ⊢ Ⅎ𝑥⦋𝐴 / 𝑦⦌𝐵 | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | nftru 1803 | . . 3 ⊢ Ⅎ𝑦⊤ | |
| 2 | nfcsb.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
| 3 | 2 | a1i 11 | . . 3 ⊢ (⊤ → Ⅎ𝑥𝐴) | 
| 4 | nfcsb.2 | . . . 4 ⊢ Ⅎ𝑥𝐵 | |
| 5 | 4 | a1i 11 | . . 3 ⊢ (⊤ → Ⅎ𝑥𝐵) | 
| 6 | 1, 3, 5 | nfcsbd 3923 | . 2 ⊢ (⊤ → Ⅎ𝑥⦋𝐴 / 𝑦⦌𝐵) | 
| 7 | 6 | mptru 1546 | 1 ⊢ Ⅎ𝑥⦋𝐴 / 𝑦⦌𝐵 | 
| Colors of variables: wff setvar class | 
| Syntax hints: ⊤wtru 1540 Ⅎwnfc 2889 ⦋csb 3898 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-13 2376 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1542 df-ex 1779 df-nf 1783 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-sbc 3788 df-csb 3899 | 
| This theorem is referenced by: cbvralcsf 3940 cbvreucsf 3942 cbvrabcsf 3943 elfvmptrab1 7043 elovmporab1 7682 | 
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