Theorem mdandyv4

Description: Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ph, ps accordingly. (Contributed by Jarvin Udandy, 6-Sep-2016)

	Ref	Expression
Hypotheses	mdandyv4.1	⊢ ( 𝜑 ↔ ⊥ )
	mdandyv4.2	⊢ ( 𝜓 ↔ ⊤ )
	mdandyv4.3	⊢ ( 𝜒 ↔ ⊥ )
	mdandyv4.4	⊢ ( 𝜃 ↔ ⊥ )
	mdandyv4.5	⊢ ( 𝜏 ↔ ⊤ )
	mdandyv4.6	⊢ ( 𝜂 ↔ ⊥ )
Assertion	mdandyv4	⊢ ( ( ( ( 𝜒 ↔ 𝜑 ) ∧ ( 𝜃 ↔ 𝜑 ) ) ∧ ( 𝜏 ↔ 𝜓 ) ) ∧ ( 𝜂 ↔ 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		mdandyv4.1	⊢ ( 𝜑 ↔ ⊥ )
2		mdandyv4.2	⊢ ( 𝜓 ↔ ⊤ )
3		mdandyv4.3	⊢ ( 𝜒 ↔ ⊥ )
4		mdandyv4.4	⊢ ( 𝜃 ↔ ⊥ )
5		mdandyv4.5	⊢ ( 𝜏 ↔ ⊤ )
6		mdandyv4.6	⊢ ( 𝜂 ↔ ⊥ )
7	3 1	bothfbothsame	⊢ ( 𝜒 ↔ 𝜑 )
8	4 1	bothfbothsame	⊢ ( 𝜃 ↔ 𝜑 )
9	7 8	pm3.2i	⊢ ( ( 𝜒 ↔ 𝜑 ) ∧ ( 𝜃 ↔ 𝜑 ) )
10	5 2	bothtbothsame	⊢ ( 𝜏 ↔ 𝜓 )
11	9 10	pm3.2i	⊢ ( ( ( 𝜒 ↔ 𝜑 ) ∧ ( 𝜃 ↔ 𝜑 ) ) ∧ ( 𝜏 ↔ 𝜓 ) )
12	6 1	bothfbothsame	⊢ ( 𝜂 ↔ 𝜑 )
13	11 12	pm3.2i	⊢ ( ( ( ( 𝜒 ↔ 𝜑 ) ∧ ( 𝜃 ↔ 𝜑 ) ) ∧ ( 𝜏 ↔ 𝜓 ) ) ∧ ( 𝜂 ↔ 𝜑 ) )