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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 528 | . 2 ⊢ (𝜑 → (𝜒 ↔ (𝜑 ∧ 𝜒))) | |
3 | 1, 2 | bitr4d 282 | 1 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 |
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 207 df-an 396 |
This theorem is referenced by: ceqsrexv 3663 raltpd 4806 opelopab2a 5558 ov 7590 ovg 7611 soseq 8196 ltprord 11095 isfull 17972 isfth 17976 sltval 27701 axcontlem5 28992 isph 30845 cmbr 31607 cvbr 32305 mdbr 32317 dmdbr 32322 brfldext 33652 brfinext 33658 risc 37894 |
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