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Mirrors > Home > ILE Home > Th. List > xorbi2d | GIF version |
Description: Deduction joining an equivalence and a left operand to form equivalence of exclusive-or. (Contributed by Jim Kingdon, 7-Oct-2018.) |
Ref | Expression |
---|---|
xorbid.1 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
xorbi2d | ⊢ (𝜑 → ((𝜃 ⊻ 𝜓) ↔ (𝜃 ⊻ 𝜒))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xorbid.1 | . . . 4 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
2 | 1 | orbi2d 736 | . . 3 ⊢ (𝜑 → ((𝜃 ∨ 𝜓) ↔ (𝜃 ∨ 𝜒))) |
3 | 1 | anbi2d 451 | . . . 4 ⊢ (𝜑 → ((𝜃 ∧ 𝜓) ↔ (𝜃 ∧ 𝜒))) |
4 | 3 | notbid 624 | . . 3 ⊢ (𝜑 → (¬ (𝜃 ∧ 𝜓) ↔ ¬ (𝜃 ∧ 𝜒))) |
5 | 2, 4 | anbi12d 456 | . 2 ⊢ (𝜑 → (((𝜃 ∨ 𝜓) ∧ ¬ (𝜃 ∧ 𝜓)) ↔ ((𝜃 ∨ 𝜒) ∧ ¬ (𝜃 ∧ 𝜒)))) |
6 | df-xor 1307 | . 2 ⊢ ((𝜃 ⊻ 𝜓) ↔ ((𝜃 ∨ 𝜓) ∧ ¬ (𝜃 ∧ 𝜓))) | |
7 | df-xor 1307 | . 2 ⊢ ((𝜃 ⊻ 𝜒) ↔ ((𝜃 ∨ 𝜒) ∧ ¬ (𝜃 ∧ 𝜒))) | |
8 | 5, 6, 7 | 3bitr4g 221 | 1 ⊢ (𝜑 → ((𝜃 ⊻ 𝜓) ↔ (𝜃 ⊻ 𝜒))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 102 ↔ wb 103 ∨ wo 661 ⊻ wxo 1306 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 |
This theorem depends on definitions: df-bi 115 df-xor 1307 |
This theorem is referenced by: xorbi12d 1313 |
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