Theorem mdandyvr4 39785

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
mdandyvr4.1	⊢ (𝜑 ↔ 𝜁)
mdandyvr4.2	⊢ (𝜓 ↔ 𝜎)
mdandyvr4.3	⊢ (𝜒 ↔ 𝜑)
mdandyvr4.4	⊢ (𝜃 ↔ 𝜑)
mdandyvr4.5	⊢ (𝜏 ↔ 𝜓)
mdandyvr4.6	⊢ (𝜂 ↔ 𝜑)

Assertion

Ref	Expression
mdandyvr4	⊢ ((((𝜒 ↔ 𝜁) ∧ (𝜃 ↔ 𝜁)) ∧ (𝜏 ↔ 𝜎)) ∧ (𝜂 ↔ 𝜁))

Proof of Theorem mdandyvr4

Step	Hyp	Ref	Expression
1		mdandyvr4.3	. . . . 5 ⊢ (𝜒 ↔ 𝜑)
2		mdandyvr4.1	. . . . 5 ⊢ (𝜑 ↔ 𝜁)
3	1, 2	bitri 263	. . . 4 ⊢ (𝜒 ↔ 𝜁)
4		mdandyvr4.4	. . . . 5 ⊢ (𝜃 ↔ 𝜑)
5	4, 2	bitri 263	. . . 4 ⊢ (𝜃 ↔ 𝜁)
6	3, 5	pm3.2i 470	. . 3 ⊢ ((𝜒 ↔ 𝜁) ∧ (𝜃 ↔ 𝜁))
7		mdandyvr4.5	. . . 4 ⊢ (𝜏 ↔ 𝜓)
8		mdandyvr4.2	. . . 4 ⊢ (𝜓 ↔ 𝜎)
9	7, 8	bitri 263	. . 3 ⊢ (𝜏 ↔ 𝜎)
10	6, 9	pm3.2i 470	. 2 ⊢ (((𝜒 ↔ 𝜁) ∧ (𝜃 ↔ 𝜁)) ∧ (𝜏 ↔ 𝜎))
11		mdandyvr4.6	. . 3 ⊢ (𝜂 ↔ 𝜑)
12	11, 2	bitri 263	. 2 ⊢ (𝜂 ↔ 𝜁)
13	10, 12	pm3.2i 470	1 ⊢ ((((𝜒 ↔ 𝜁) ∧ (𝜃 ↔ 𝜁)) ∧ (𝜏 ↔ 𝜎)) ∧ (𝜂 ↔ 𝜁))

Syntax hints:   ↔ wb 195   ∧ wa 383

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 196  df-an 385

This theorem is referenced by:  mdandyvr11  39792