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Mirrors > Home > ILE Home > Th. List > 2thd | GIF version |
Description: Two truths are equivalent (deduction rule). (Contributed by NM, 3-Jun-2012.) (Revised by NM, 29-Jan-2013.) |
Ref | Expression |
---|---|
2thd.1 | ⊢ (𝜑 → 𝜓) |
2thd.2 | ⊢ (𝜑 → 𝜒) |
Ref | Expression |
---|---|
2thd | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2thd.1 | . 2 ⊢ (𝜑 → 𝜓) | |
2 | 2thd.2 | . 2 ⊢ (𝜑 → 𝜒) | |
3 | pm5.1im 162 | . 2 ⊢ (𝜓 → (𝜒 → (𝜓 ↔ 𝜒))) | |
4 | 1, 2, 3 | sylc 56 | 1 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia2 100 ax-ia3 101 |
This theorem depends on definitions: df-bi 110 |
This theorem is referenced by: abvor0dc 3242 euotd 3991 nn0eln0 4341 elabrex 5397 riota5f 5492 nntri3 6075 fin0 6342 nn1m1nn 7932 xrlttri3 8718 nltpnft 8730 ngtmnft 8731 xrrebnd 8732 iccshftr 8862 iccshftl 8864 iccdil 8866 icccntr 8868 fzaddel 8922 elfzomelpfzo 9087 nnesq 9368 |
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