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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 6567 | . . 3 ⊢ (𝜑 → (𝐵 − 𝐷) = (𝐶 − 𝐸)) |
4 | 3 | breq2d 4595 | . 2 ⊢ (𝜑 → (𝐴 ∥ (𝐵 − 𝐷) ↔ 𝐴 ∥ (𝐶 − 𝐸))) |
5 | 2 | negeqd 10154 | . . . 4 ⊢ (𝜑 → -𝐷 = -𝐸) |
6 | 1, 5 | oveq12d 6567 | . . 3 ⊢ (𝜑 → (𝐵 − -𝐷) = (𝐶 − -𝐸)) |
7 | 6 | breq2d 4595 | . 2 ⊢ (𝜑 → (𝐴 ∥ (𝐵 − -𝐷) ↔ 𝐴 ∥ (𝐶 − -𝐸))) |
8 | 4, 7 | orbi12d 742 | 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 195 ∨ wo 382 = wceq 1475 class class class wbr 4583 (class class class)co 6549 − cmin 10145 -cneg 10146 ∥ cdvds 14821 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-iota 5768 df-fv 5812 df-ov 6552 df-neg 10148 |
This theorem is referenced by: acongrep 36565 jm2.26a 36585 jm2.26 36587 |
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