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Mirrors > Home > MPE Home > Th. List > nfbid | Structured version Visualization version GIF version |
Description: If in a context 𝑥 is not free in 𝜓 and 𝜒, then it is not free in (𝜓 ↔ 𝜒). (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 29-Dec-2017.) |
Ref | Expression |
---|---|
nfbid.1 | ⊢ (𝜑 → Ⅎ𝑥𝜓) |
nfbid.2 | ⊢ (𝜑 → Ⅎ𝑥𝜒) |
Ref | Expression |
---|---|
nfbid | ⊢ (𝜑 → Ⅎ𝑥(𝜓 ↔ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfbi2 477 | . 2 ⊢ ((𝜓 ↔ 𝜒) ↔ ((𝜓 → 𝜒) ∧ (𝜒 → 𝜓))) | |
2 | nfbid.1 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
3 | nfbid.2 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜒) | |
4 | 2, 3 | nfimd 1895 | . . 3 ⊢ (𝜑 → Ⅎ𝑥(𝜓 → 𝜒)) |
5 | 3, 2 | nfimd 1895 | . . 3 ⊢ (𝜑 → Ⅎ𝑥(𝜒 → 𝜓)) |
6 | 4, 5 | nfand 1898 | . 2 ⊢ (𝜑 → Ⅎ𝑥((𝜓 → 𝜒) ∧ (𝜒 → 𝜓))) |
7 | 1, 6 | nfxfrd 1854 | 1 ⊢ (𝜑 → Ⅎ𝑥(𝜓 ↔ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 Ⅎwnf 1784 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ex 1781 df-nf 1785 |
This theorem is referenced by: nfbi 1904 nfeqd 2988 nfiotadw 6317 nfiotad 6319 iota2df 6342 axextnd 10013 axrepndlem1 10014 axrepndlem2 10015 axacndlem4 10032 axacndlem5 10033 axacnd 10034 axextdist 33044 copsex2d 34434 cbveud 34656 wl-eudf 34823 wl-sb8eut 34828 |
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