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Metamath Proof Explorer

Theorem caovord3

Description: Ordering law. (Contributed by NM, 29-Feb-1996)

Ref Expression
Hypotheses caovord.1 ⊢ 𝐴 ∈ V
caovord.2 ⊢ 𝐵 ∈ V
caovord.3 ⊢ ( 𝑧 ∈ 𝑆 → ( 𝑥 𝑅 𝑦 ↔ ( 𝑧 𝐹 𝑥 ) 𝑅 ( 𝑧 𝐹 𝑦 ) ) )
caovord2.3 ⊢ 𝐶 ∈ V
caovord2.com ⊢ ( 𝑥 𝐹 𝑦 ) = ( 𝑦 𝐹 𝑥 )
caovord3.4 ⊢ 𝐷 ∈ V
Assertion caovord3 ( ( ( 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ) ∧ ( 𝐴 𝐹 𝐵 ) = ( 𝐶 𝐹 𝐷 ) ) → ( 𝐴 𝑅 𝐶 ↔ 𝐷 𝑅 𝐵 ) )

Proof

Step Hyp Ref Expression
1 caovord.1 ⊢ 𝐴 ∈ V
2 caovord.2 ⊢ 𝐵 ∈ V
3 caovord.3 ⊢ ( 𝑧 ∈ 𝑆 → ( 𝑥 𝑅 𝑦 ↔ ( 𝑧 𝐹 𝑥 ) 𝑅 ( 𝑧 𝐹 𝑦 ) ) )
4 caovord2.3 ⊢ 𝐶 ∈ V
5 caovord2.com ⊢ ( 𝑥 𝐹 𝑦 ) = ( 𝑦 𝐹 𝑥 )
6 caovord3.4 ⊢ 𝐷 ∈ V
7 1 4 3 2 5 caovord2 ⊢ ( 𝐵 ∈ 𝑆 → ( 𝐴 𝑅 𝐶 ↔ ( 𝐴 𝐹 𝐵 ) 𝑅 ( 𝐶 𝐹 𝐵 ) ) )
8 7 adantr ⊢ ( ( 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ) → ( 𝐴 𝑅 𝐶 ↔ ( 𝐴 𝐹 𝐵 ) 𝑅 ( 𝐶 𝐹 𝐵 ) ) )
9 breq1 ⊢ ( ( 𝐴 𝐹 𝐵 ) = ( 𝐶 𝐹 𝐷 ) → ( ( 𝐴 𝐹 𝐵 ) 𝑅 ( 𝐶 𝐹 𝐵 ) ↔ ( 𝐶 𝐹 𝐷 ) 𝑅 ( 𝐶 𝐹 𝐵 ) ) )
10 8 9 sylan9bb ⊢ ( ( ( 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ) ∧ ( 𝐴 𝐹 𝐵 ) = ( 𝐶 𝐹 𝐷 ) ) → ( 𝐴 𝑅 𝐶 ↔ ( 𝐶 𝐹 𝐷 ) 𝑅 ( 𝐶 𝐹 𝐵 ) ) )
11 6 2 3 caovord ⊢ ( 𝐶 ∈ 𝑆 → ( 𝐷 𝑅 𝐵 ↔ ( 𝐶 𝐹 𝐷 ) 𝑅 ( 𝐶 𝐹 𝐵 ) ) )
12 11 ad2antlr ⊢ ( ( ( 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ) ∧ ( 𝐴 𝐹 𝐵 ) = ( 𝐶 𝐹 𝐷 ) ) → ( 𝐷 𝑅 𝐵 ↔ ( 𝐶 𝐹 𝐷 ) 𝑅 ( 𝐶 𝐹 𝐵 ) ) )
13 10 12 bitr4d ⊢ ( ( ( 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ) ∧ ( 𝐴 𝐹 𝐵 ) = ( 𝐶 𝐹 𝐷 ) ) → ( 𝐴 𝑅 𝐶 ↔ 𝐷 𝑅 𝐵 ) )