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Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > acongeq12d | Structured version Visualization version GIF version |
Description: Substitution deduction for alternating congruence. (Contributed by Stefan O'Rear, 3-Oct-2014.) |
Ref | Expression |
---|---|
acongeq12d.1 | ⊢ (𝜑 → 𝐵 = 𝐶) |
acongeq12d.2 | ⊢ (𝜑 → 𝐷 = 𝐸) |
Ref | Expression |
---|---|
acongeq12d | ⊢ (𝜑 → ((𝐴 ∥ (𝐵 − 𝐷) ∨ 𝐴 ∥ (𝐵 − -𝐷)) ↔ (𝐴 ∥ (𝐶 − 𝐸) ∨ 𝐴 ∥ (𝐶 − -𝐸)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | acongeq12d.1 | . . . 4 ⊢ (𝜑 → 𝐵 = 𝐶) | |
2 | acongeq12d.2 | . . . 4 ⊢ (𝜑 → 𝐷 = 𝐸) | |
3 | 1, 2 | oveq12d 7376 | . . 3 ⊢ (𝜑 → (𝐵 − 𝐷) = (𝐶 − 𝐸)) |
4 | 3 | breq2d 5118 | . 2 ⊢ (𝜑 → (𝐴 ∥ (𝐵 − 𝐷) ↔ 𝐴 ∥ (𝐶 − 𝐸))) |
5 | 2 | negeqd 11400 | . . . 4 ⊢ (𝜑 → -𝐷 = -𝐸) |
6 | 1, 5 | oveq12d 7376 | . . 3 ⊢ (𝜑 → (𝐵 − -𝐷) = (𝐶 − -𝐸)) |
7 | 6 | breq2d 5118 | . 2 ⊢ (𝜑 → (𝐴 ∥ (𝐵 − -𝐷) ↔ 𝐴 ∥ (𝐶 − -𝐸))) |
8 | 4, 7 | orbi12d 918 | 1 ⊢ (𝜑 → ((𝐴 ∥ (𝐵 − 𝐷) ∨ 𝐴 ∥ (𝐵 − -𝐷)) ↔ (𝐴 ∥ (𝐶 − 𝐸) ∨ 𝐴 ∥ (𝐶 − -𝐸)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∨ wo 846 = wceq 1542 class class class wbr 5106 (class class class)co 7358 − cmin 11390 -cneg 11391 ∥ cdvds 16141 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-iota 6449 df-fv 6505 df-ov 7361 df-neg 11393 |
This theorem is referenced by: acongrep 41347 jm2.26a 41367 jm2.26 41369 |
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