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Mirrors > Home > ILE Home > Th. List > xorbi1d | GIF version |
Description: Deduction joining an equivalence and a right operand to form equivalence of exclusive-or. (Contributed by Jim Kingdon, 7-Oct-2018.) |
Ref | Expression |
---|---|
xorbid.1 | ⊢ (φ → (ψ ↔ χ)) |
Ref | Expression |
---|---|
xorbi1d | ⊢ (φ → ((ψ ⊻ θ) ↔ (χ ⊻ θ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xorbid.1 | . . . 4 ⊢ (φ → (ψ ↔ χ)) | |
2 | 1 | orbi1d 704 | . . 3 ⊢ (φ → ((ψ ∨ θ) ↔ (χ ∨ θ))) |
3 | 1 | anbi1d 438 | . . . 4 ⊢ (φ → ((ψ ∧ θ) ↔ (χ ∧ θ))) |
4 | 3 | notbid 591 | . . 3 ⊢ (φ → (¬ (ψ ∧ θ) ↔ ¬ (χ ∧ θ))) |
5 | 2, 4 | anbi12d 442 | . 2 ⊢ (φ → (((ψ ∨ θ) ∧ ¬ (ψ ∧ θ)) ↔ ((χ ∨ θ) ∧ ¬ (χ ∧ θ)))) |
6 | df-xor 1266 | . 2 ⊢ ((ψ ⊻ θ) ↔ ((ψ ∨ θ) ∧ ¬ (ψ ∧ θ))) | |
7 | df-xor 1266 | . 2 ⊢ ((χ ⊻ θ) ↔ ((χ ∨ θ) ∧ ¬ (χ ∧ θ))) | |
8 | 5, 6, 7 | 3bitr4g 212 | 1 ⊢ (φ → ((ψ ⊻ θ) ↔ (χ ⊻ θ))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 97 ↔ wb 98 ∨ wo 628 ⊻ wxo 1265 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 629 |
This theorem depends on definitions: df-bi 110 df-xor 1266 |
This theorem is referenced by: xorbi12d 1271 |
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