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Mirrors > Home > MPE Home > Th. List > bianabs | Structured version Visualization version GIF version |
Description: Absorb a hypothesis into the second member of a biconditional. (Contributed by FL, 15-Feb-2007.) |
Ref | Expression |
---|---|
bianabs.1 | ⊢ (𝜑 → (𝜓 ↔ (𝜑 ∧ 𝜒))) |
Ref | Expression |
---|---|
bianabs | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bianabs.1 | . 2 ⊢ (𝜑 → (𝜓 ↔ (𝜑 ∧ 𝜒))) | |
2 | ibar 529 | . 2 ⊢ (𝜑 → (𝜒 ↔ (𝜑 ∧ 𝜒))) | |
3 | 1, 2 | bitr4d 281 | 1 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 206 df-an 397 |
This theorem is referenced by: ceqsrexv 3605 raltpd 4742 opelopab2a 5492 ov 7498 ovg 7518 soseq 8090 ltprord 10965 isfull 17796 isfth 17800 sltval 26993 axcontlem5 27915 isph 29762 cmbr 30524 cvbr 31222 mdbr 31234 dmdbr 31239 brfldext 32327 brfinext 32333 risc 36436 |
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