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Mirrors > Home > ILE Home > Th. List > 2falsed | GIF version |
Description: Two falsehoods are equivalent (deduction form). (Contributed by NM, 11-Oct-2013.) |
Ref | Expression |
---|---|
2falsed.1 | ⊢ (𝜑 → ¬ 𝜓) |
2falsed.2 | ⊢ (𝜑 → ¬ 𝜒) |
Ref | Expression |
---|---|
2falsed | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2falsed.1 | . . 3 ⊢ (𝜑 → ¬ 𝜓) | |
2 | 1 | pm2.21d 619 | . 2 ⊢ (𝜑 → (𝜓 → 𝜒)) |
3 | 2falsed.2 | . . 3 ⊢ (𝜑 → ¬ 𝜒) | |
4 | 3 | pm2.21d 619 | . 2 ⊢ (𝜑 → (𝜒 → 𝜓)) |
5 | 2, 4 | impbid 129 | 1 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 105 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia2 107 ax-ia3 108 ax-in2 615 |
This theorem depends on definitions: df-bi 117 |
This theorem is referenced by: pm5.21ni 703 bianfd 948 abvor0dc 3448 nn0eln0 4621 nntri3 6500 fin0 6887 omp1eomlem 7095 ctssdccl 7112 ismkvnex 7155 xrlttri3 9799 nltpnft 9816 ngtmnft 9819 xrrebnd 9821 xltadd1 9878 xposdif 9884 xleaddadd 9889 hashnncl 10777 zfz1isolemiso 10821 mod2eq1n2dvds 11886 m1exp1 11908 pceq0 12323 |
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