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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 6668 | . . 3 ⊢ (𝜑 → (𝐵 − 𝐷) = (𝐶 − 𝐸)) |
4 | 3 | breq2d 4665 | . 2 ⊢ (𝜑 → (𝐴 ∥ (𝐵 − 𝐷) ↔ 𝐴 ∥ (𝐶 − 𝐸))) |
5 | 2 | negeqd 10275 | . . . 4 ⊢ (𝜑 → -𝐷 = -𝐸) |
6 | 1, 5 | oveq12d 6668 | . . 3 ⊢ (𝜑 → (𝐵 − -𝐷) = (𝐶 − -𝐸)) |
7 | 6 | breq2d 4665 | . 2 ⊢ (𝜑 → (𝐴 ∥ (𝐵 − -𝐷) ↔ 𝐴 ∥ (𝐶 − -𝐸))) |
8 | 4, 7 | orbi12d 746 | 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 196 ∨ wo 383 = wceq 1483 class class class wbr 4653 (class class class)co 6650 − cmin 10266 -cneg 10267 ∥ cdvds 14983 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-iota 5851 df-fv 5896 df-ov 6653 df-neg 10269 |
This theorem is referenced by: acongrep 37547 jm2.26a 37567 jm2.26 37569 |
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