Theorem mdandyvr2 44349

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
mdandyvr2.1	⊢ (𝜑 ↔ 𝜁)
mdandyvr2.2	⊢ (𝜓 ↔ 𝜎)
mdandyvr2.3	⊢ (𝜒 ↔ 𝜑)
mdandyvr2.4	⊢ (𝜃 ↔ 𝜓)
mdandyvr2.5	⊢ (𝜏 ↔ 𝜑)
mdandyvr2.6	⊢ (𝜂 ↔ 𝜑)

Assertion

Ref	Expression
mdandyvr2	⊢ ((((𝜒 ↔ 𝜁) ∧ (𝜃 ↔ 𝜎)) ∧ (𝜏 ↔ 𝜁)) ∧ (𝜂 ↔ 𝜁))

Proof of Theorem mdandyvr2

Step	Hyp	Ref	Expression
1		mdandyvr2.3	. . . . 5 ⊢ (𝜒 ↔ 𝜑)
2		mdandyvr2.1	. . . . 5 ⊢ (𝜑 ↔ 𝜁)
3	1, 2	bitri 274	. . . 4 ⊢ (𝜒 ↔ 𝜁)
4		mdandyvr2.4	. . . . 5 ⊢ (𝜃 ↔ 𝜓)
5		mdandyvr2.2	. . . . 5 ⊢ (𝜓 ↔ 𝜎)
6	4, 5	bitri 274	. . . 4 ⊢ (𝜃 ↔ 𝜎)
7	3, 6	pm3.2i 470	. . 3 ⊢ ((𝜒 ↔ 𝜁) ∧ (𝜃 ↔ 𝜎))
8		mdandyvr2.5	. . . 4 ⊢ (𝜏 ↔ 𝜑)
9	8, 2	bitri 274	. . 3 ⊢ (𝜏 ↔ 𝜁)
10	7, 9	pm3.2i 470	. 2 ⊢ (((𝜒 ↔ 𝜁) ∧ (𝜃 ↔ 𝜎)) ∧ (𝜏 ↔ 𝜁))
11		mdandyvr2.6	. . 3 ⊢ (𝜂 ↔ 𝜑)
12	11, 2	bitri 274	. 2 ⊢ (𝜂 ↔ 𝜁)
13	10, 12	pm3.2i 470	1 ⊢ ((((𝜒 ↔ 𝜁) ∧ (𝜃 ↔ 𝜎)) ∧ (𝜏 ↔ 𝜁)) ∧ (𝜂 ↔ 𝜁))

Syntax hints:   ↔ wb 205   ∧ wa 395

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 206  df-an 396

This theorem is referenced by:  mdandyvr13  44360