Mathbox for Stefan O'Rear |
< Previous
Next >
Nearby theorems |
||
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 7155 | . . 3 ⊢ (𝜑 → (𝐵 − 𝐷) = (𝐶 − 𝐸)) |
4 | 3 | breq2d 5059 | . 2 ⊢ (𝜑 → (𝐴 ∥ (𝐵 − 𝐷) ↔ 𝐴 ∥ (𝐶 − 𝐸))) |
5 | 2 | negeqd 10861 | . . . 4 ⊢ (𝜑 → -𝐷 = -𝐸) |
6 | 1, 5 | oveq12d 7155 | . . 3 ⊢ (𝜑 → (𝐵 − -𝐷) = (𝐶 − -𝐸)) |
7 | 6 | breq2d 5059 | . 2 ⊢ (𝜑 → (𝐴 ∥ (𝐵 − -𝐷) ↔ 𝐴 ∥ (𝐶 − -𝐸))) |
8 | 4, 7 | orbi12d 915 | 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 208 ∨ wo 843 = wceq 1537 class class class wbr 5047 (class class class)co 7137 − cmin 10851 -cneg 10852 ∥ cdvds 15587 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2792 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2799 df-cleq 2813 df-clel 2891 df-nfc 2959 df-rab 3142 df-v 3483 df-dif 3922 df-un 3924 df-in 3926 df-ss 3935 df-nul 4275 df-if 4449 df-sn 4549 df-pr 4551 df-op 4555 df-uni 4820 df-br 5048 df-iota 6295 df-fv 6344 df-ov 7140 df-neg 10854 |
This theorem is referenced by: acongrep 39664 jm2.26a 39684 jm2.26 39686 |
Copyright terms: Public domain | W3C validator |