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Mirrors > Home > ILE Home > Th. List > nf3an | GIF version |
Description: If 𝑥 is not free in 𝜑, 𝜓, and 𝜒, it is not free in (𝜑 ∧ 𝜓 ∧ 𝜒). (Contributed by Mario Carneiro, 11-Aug-2016.) |
Ref | Expression |
---|---|
nfan.1 | ⊢ Ⅎ𝑥𝜑 |
nfan.2 | ⊢ Ⅎ𝑥𝜓 |
nfan.3 | ⊢ Ⅎ𝑥𝜒 |
Ref | Expression |
---|---|
nf3an | ⊢ Ⅎ𝑥(𝜑 ∧ 𝜓 ∧ 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-3an 969 | . 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∧ 𝜒)) | |
2 | nfan.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
3 | nfan.2 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
4 | 2, 3 | nfan 1552 | . . 3 ⊢ Ⅎ𝑥(𝜑 ∧ 𝜓) |
5 | nfan.3 | . . 3 ⊢ Ⅎ𝑥𝜒 | |
6 | 4, 5 | nfan 1552 | . 2 ⊢ Ⅎ𝑥((𝜑 ∧ 𝜓) ∧ 𝜒) |
7 | 1, 6 | nfxfr 1461 | 1 ⊢ Ⅎ𝑥(𝜑 ∧ 𝜓 ∧ 𝜒) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ∧ w3a 967 Ⅎwnf 1447 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1434 ax-gen 1436 ax-4 1497 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-nf 1448 |
This theorem is referenced by: vtocl3gaf 2790 mob 2903 nfop 3768 mkvprop 7113 seq3f1olemstep 10426 seq3f1olemp 10427 nfsum1 11283 nfsum 11284 |
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