Theorem mdandyvr1

Description: Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ze, si accordingly. (Contributed by Jarvin Udandy, 7-Sep-2016)

	Ref	Expression
Hypotheses	mdandyvr1.1	⊢ ( 𝜑 ↔ 𝜁 )
	mdandyvr1.2	⊢ ( 𝜓 ↔ 𝜎 )
	mdandyvr1.3	⊢ ( 𝜒 ↔ 𝜓 )
	mdandyvr1.4	⊢ ( 𝜃 ↔ 𝜑 )
	mdandyvr1.5	⊢ ( 𝜏 ↔ 𝜑 )
	mdandyvr1.6	⊢ ( 𝜂 ↔ 𝜑 )
Assertion	mdandyvr1	⊢ ( ( ( ( 𝜒 ↔ 𝜎 ) ∧ ( 𝜃 ↔ 𝜁 ) ) ∧ ( 𝜏 ↔ 𝜁 ) ) ∧ ( 𝜂 ↔ 𝜁 ) )

Proof

Step	Hyp	Ref	Expression
1		mdandyvr1.1	⊢ ( 𝜑 ↔ 𝜁 )
2		mdandyvr1.2	⊢ ( 𝜓 ↔ 𝜎 )
3		mdandyvr1.3	⊢ ( 𝜒 ↔ 𝜓 )
4		mdandyvr1.4	⊢ ( 𝜃 ↔ 𝜑 )
5		mdandyvr1.5	⊢ ( 𝜏 ↔ 𝜑 )
6		mdandyvr1.6	⊢ ( 𝜂 ↔ 𝜑 )
7	3 2	bitri	⊢ ( 𝜒 ↔ 𝜎 )
8	4 1	bitri	⊢ ( 𝜃 ↔ 𝜁 )
9	7 8	pm3.2i	⊢ ( ( 𝜒 ↔ 𝜎 ) ∧ ( 𝜃 ↔ 𝜁 ) )
10	5 1	bitri	⊢ ( 𝜏 ↔ 𝜁 )
11	9 10	pm3.2i	⊢ ( ( ( 𝜒 ↔ 𝜎 ) ∧ ( 𝜃 ↔ 𝜁 ) ) ∧ ( 𝜏 ↔ 𝜁 ) )
12	6 1	bitri	⊢ ( 𝜂 ↔ 𝜁 )
13	11 12	pm3.2i	⊢ ( ( ( ( 𝜒 ↔ 𝜎 ) ∧ ( 𝜃 ↔ 𝜁 ) ) ∧ ( 𝜏 ↔ 𝜁 ) ) ∧ ( 𝜂 ↔ 𝜁 ) )