Theorem mdandyvr7 47964

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
mdandyvr7.1	⊢ (𝜑 ↔ 𝜁)
mdandyvr7.2	⊢ (𝜓 ↔ 𝜎)
mdandyvr7.3	⊢ (𝜒 ↔ 𝜓)
mdandyvr7.4	⊢ (𝜃 ↔ 𝜓)
mdandyvr7.5	⊢ (𝜏 ↔ 𝜓)
mdandyvr7.6	⊢ (𝜂 ↔ 𝜑)

Assertion

Ref	Expression
mdandyvr7	⊢ ((((𝜒 ↔ 𝜎) ∧ (𝜃 ↔ 𝜎)) ∧ (𝜏 ↔ 𝜎)) ∧ (𝜂 ↔ 𝜁))

Proof of Theorem mdandyvr7

Step	Hyp	Ref	Expression
1		mdandyvr7.3	. . . . 5 ⊢ (𝜒 ↔ 𝜓)
2		mdandyvr7.2	. . . . 5 ⊢ (𝜓 ↔ 𝜎)
3	1, 2	bitri 278	. . . 4 ⊢ (𝜒 ↔ 𝜎)
4		mdandyvr7.4	. . . . 5 ⊢ (𝜃 ↔ 𝜓)
5	4, 2	bitri 278	. . . 4 ⊢ (𝜃 ↔ 𝜎)
6	3, 5	pm3.2i 476	. . 3 ⊢ ((𝜒 ↔ 𝜎) ∧ (𝜃 ↔ 𝜎))
7		mdandyvr7.5	. . . 4 ⊢ (𝜏 ↔ 𝜓)
8	7, 2	bitri 278	. . 3 ⊢ (𝜏 ↔ 𝜎)
9	6, 8	pm3.2i 476	. 2 ⊢ (((𝜒 ↔ 𝜎) ∧ (𝜃 ↔ 𝜎)) ∧ (𝜏 ↔ 𝜎))
10		mdandyvr7.6	. . 3 ⊢ (𝜂 ↔ 𝜑)
11		mdandyvr7.1	. . 3 ⊢ (𝜑 ↔ 𝜁)
12	10, 11	bitri 278	. 2 ⊢ (𝜂 ↔ 𝜁)
13	9, 12	pm3.2i 476	1 ⊢ ((((𝜒 ↔ 𝜎) ∧ (𝜃 ↔ 𝜎)) ∧ (𝜏 ↔ 𝜎)) ∧ (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:  mdandyvr8  47965