Theorem mdandyvr1 47958

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.)

Hypotheses

Ref	Expression
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

Ref	Expression
mdandyvr1	⊢ ((((𝜒 ↔ 𝜎) ∧ (𝜃 ↔ 𝜁)) ∧ (𝜏 ↔ 𝜁)) ∧ (𝜂 ↔ 𝜁))

Proof of Theorem mdandyvr1

Step	Hyp	Ref	Expression
1		mdandyvr1.3	. . . . 5 ⊢ (𝜒 ↔ 𝜓)
2		mdandyvr1.2	. . . . 5 ⊢ (𝜓 ↔ 𝜎)
3	1, 2	bitri 278	. . . 4 ⊢ (𝜒 ↔ 𝜎)
4		mdandyvr1.4	. . . . 5 ⊢ (𝜃 ↔ 𝜑)
5		mdandyvr1.1	. . . . 5 ⊢ (𝜑 ↔ 𝜁)
6	4, 5	bitri 278	. . . 4 ⊢ (𝜃 ↔ 𝜁)
7	3, 6	pm3.2i 476	. . 3 ⊢ ((𝜒 ↔ 𝜎) ∧ (𝜃 ↔ 𝜁))
8		mdandyvr1.5	. . . 4 ⊢ (𝜏 ↔ 𝜑)
9	8, 5	bitri 278	. . 3 ⊢ (𝜏 ↔ 𝜁)
10	7, 9	pm3.2i 476	. 2 ⊢ (((𝜒 ↔ 𝜎) ∧ (𝜃 ↔ 𝜁)) ∧ (𝜏 ↔ 𝜁))
11		mdandyvr1.6	. . 3 ⊢ (𝜂 ↔ 𝜑)
12	11, 5	bitri 278	. 2 ⊢ (𝜂 ↔ 𝜁)
13	10, 12	pm3.2i 476	1 ⊢ ((((𝜒 ↔ 𝜎) ∧ (𝜃 ↔ 𝜁)) ∧ (𝜏 ↔ 𝜁)) ∧ (𝜂 ↔ 𝜁))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402

This theorem is used by:  mdandyvr14  47971