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Mirrors > Home > ILE Home > Th. List > 3bitr3g | GIF version |
Description: More general version of 3bitr3i 209. Useful for converting definitions in a formula. (Contributed by NM, 4-Jun-1995.) |
Ref | Expression |
---|---|
3bitr3g.1 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
3bitr3g.2 | ⊢ (𝜓 ↔ 𝜃) |
3bitr3g.3 | ⊢ (𝜒 ↔ 𝜏) |
Ref | Expression |
---|---|
3bitr3g | ⊢ (𝜑 → (𝜃 ↔ 𝜏)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3bitr3g.2 | . . 3 ⊢ (𝜓 ↔ 𝜃) | |
2 | 3bitr3g.1 | . . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
3 | 1, 2 | bitr3id 193 | . 2 ⊢ (𝜑 → (𝜃 ↔ 𝜒)) |
4 | 3bitr3g.3 | . 2 ⊢ (𝜒 ↔ 𝜏) | |
5 | 3, 4 | syl6bb 195 | 1 ⊢ (𝜑 → (𝜃 ↔ 𝜏)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: con2bidc 861 sbal1yz 1977 sbal1 1978 dfsbcq2 2916 iindif2m 3888 opeqex 4179 rabxfrd 4398 eqbrrdv 4644 eqbrrdiv 4645 opelco2g 4715 opelcnvg 4727 ralrnmpt 5570 rexrnmpt 5571 fliftcnv 5704 eusvobj2 5768 f1od2 6140 ottposg 6160 ercnv 6458 exmidpw 6810 djuf1olem 6946 fzen 9854 fihasheq0 10572 divalgb 11658 isprm3 11835 eldvap 12859 |
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