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Mirrors > Home > MPE Home > Th. List > Mathboxes > axorbciffatcxorb | Structured version Visualization version GIF version |
Description: Given a is equivalent to (not b), c is equivalent to a. there exists a proof for ( c xor b ) . (Contributed by Jarvin Udandy, 7-Sep-2016.) |
Ref | Expression |
---|---|
axorbciffatcxorb.1 | ⊢ (𝜑 ⊻ 𝜓) |
axorbciffatcxorb.2 | ⊢ (𝜒 ↔ 𝜑) |
Ref | Expression |
---|---|
axorbciffatcxorb | ⊢ (𝜒 ⊻ 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | axorbciffatcxorb.1 | . . . . 5 ⊢ (𝜑 ⊻ 𝜓) | |
2 | 1 | axorbtnotaiffb 41070 | . . . 4 ⊢ ¬ (𝜑 ↔ 𝜓) |
3 | xor3 372 | . . . 4 ⊢ (¬ (𝜑 ↔ 𝜓) ↔ (𝜑 ↔ ¬ 𝜓)) | |
4 | 2, 3 | mpbi 220 | . . 3 ⊢ (𝜑 ↔ ¬ 𝜓) |
5 | axorbciffatcxorb.2 | . . 3 ⊢ (𝜒 ↔ 𝜑) | |
6 | 4, 5 | aiffnbandciffatnotciffb 41071 | . 2 ⊢ ¬ (𝜒 ↔ 𝜓) |
7 | df-xor 1465 | . 2 ⊢ ((𝜒 ⊻ 𝜓) ↔ ¬ (𝜒 ↔ 𝜓)) | |
8 | 6, 7 | mpbir 221 | 1 ⊢ (𝜒 ⊻ 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 196 ⊻ wxo 1464 |
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 197 df-xor 1465 |
This theorem is referenced by: mdandyvrx0 41148 mdandyvrx1 41149 mdandyvrx2 41150 mdandyvrx3 41151 mdandyvrx4 41152 mdandyvrx5 41153 mdandyvrx6 41154 mdandyvrx7 41155 |
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