Theorem mdandyvr3 43198

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
mdandyvr3.1	⊢ (𝜑 ↔ 𝜁)
mdandyvr3.2	⊢ (𝜓 ↔ 𝜎)
mdandyvr3.3	⊢ (𝜒 ↔ 𝜓)
mdandyvr3.4	⊢ (𝜃 ↔ 𝜓)
mdandyvr3.5	⊢ (𝜏 ↔ 𝜑)
mdandyvr3.6	⊢ (𝜂 ↔ 𝜑)

Assertion

Ref	Expression
mdandyvr3	⊢ ((((𝜒 ↔ 𝜎) ∧ (𝜃 ↔ 𝜎)) ∧ (𝜏 ↔ 𝜁)) ∧ (𝜂 ↔ 𝜁))

Proof of Theorem mdandyvr3

Step	Hyp	Ref	Expression
1		mdandyvr3.3	. . . . 5 ⊢ (𝜒 ↔ 𝜓)
2		mdandyvr3.2	. . . . 5 ⊢ (𝜓 ↔ 𝜎)
3	1, 2	bitri 277	. . . 4 ⊢ (𝜒 ↔ 𝜎)
4		mdandyvr3.4	. . . . 5 ⊢ (𝜃 ↔ 𝜓)
5	4, 2	bitri 277	. . . 4 ⊢ (𝜃 ↔ 𝜎)
6	3, 5	pm3.2i 473	. . 3 ⊢ ((𝜒 ↔ 𝜎) ∧ (𝜃 ↔ 𝜎))
7		mdandyvr3.5	. . . 4 ⊢ (𝜏 ↔ 𝜑)
8		mdandyvr3.1	. . . 4 ⊢ (𝜑 ↔ 𝜁)
9	7, 8	bitri 277	. . 3 ⊢ (𝜏 ↔ 𝜁)
10	6, 9	pm3.2i 473	. 2 ⊢ (((𝜒 ↔ 𝜎) ∧ (𝜃 ↔ 𝜎)) ∧ (𝜏 ↔ 𝜁))
11		mdandyvr3.6	. . 3 ⊢ (𝜂 ↔ 𝜑)
12	11, 8	bitri 277	. 2 ⊢ (𝜂 ↔ 𝜁)
13	10, 12	pm3.2i 473	1 ⊢ ((((𝜒 ↔ 𝜎) ∧ (𝜃 ↔ 𝜎)) ∧ (𝜏 ↔ 𝜁)) ∧ (𝜂 ↔ 𝜁))

Syntax hints:   ↔ wb 208   ∧ wa 398

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 209  df-an 399

This theorem is referenced by:  mdandyvr12  43207