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Mirrors > Home > ILE Home > Th. List > bianabs | 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 285 | . 2 ⊢ (𝜑 → (𝜒 ↔ (𝜑 ∧ 𝜒))) | |
3 | 1, 2 | bitr4d 180 | 1 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 |
This theorem depends on definitions: df-bi 110 |
This theorem is referenced by: ceqsrexv 2674 opelopab2a 4002 ov 5620 ovg 5639 ltresr 6915 |
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