Theorem mdandyv7 44451

Description: Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ph, ps accordingly. (Contributed by Jarvin Udandy, 6-Sep-2016.)

Hypotheses

Ref	Expression
mdandyv7.1	⊢ (𝜑 ↔ ⊥)
mdandyv7.2	⊢ (𝜓 ↔ ⊤)
mdandyv7.3	⊢ (𝜒 ↔ ⊤)
mdandyv7.4	⊢ (𝜃 ↔ ⊤)
mdandyv7.5	⊢ (𝜏 ↔ ⊤)
mdandyv7.6	⊢ (𝜂 ↔ ⊥)

Assertion

Ref	Expression
mdandyv7	⊢ ((((𝜒 ↔ 𝜓) ∧ (𝜃 ↔ 𝜓)) ∧ (𝜏 ↔ 𝜓)) ∧ (𝜂 ↔ 𝜑))

Proof of Theorem mdandyv7

Step	Hyp	Ref	Expression
1		mdandyv7.3	. . . . 5 ⊢ (𝜒 ↔ ⊤)
2		mdandyv7.2	. . . . 5 ⊢ (𝜓 ↔ ⊤)
3	1, 2	bothtbothsame 44394	. . . 4 ⊢ (𝜒 ↔ 𝜓)
4		mdandyv7.4	. . . . 5 ⊢ (𝜃 ↔ ⊤)
5	4, 2	bothtbothsame 44394	. . . 4 ⊢ (𝜃 ↔ 𝜓)
6	3, 5	pm3.2i 471	. . 3 ⊢ ((𝜒 ↔ 𝜓) ∧ (𝜃 ↔ 𝜓))
7		mdandyv7.5	. . . 4 ⊢ (𝜏 ↔ ⊤)
8	7, 2	bothtbothsame 44394	. . 3 ⊢ (𝜏 ↔ 𝜓)
9	6, 8	pm3.2i 471	. 2 ⊢ (((𝜒 ↔ 𝜓) ∧ (𝜃 ↔ 𝜓)) ∧ (𝜏 ↔ 𝜓))
10		mdandyv7.6	. . 3 ⊢ (𝜂 ↔ ⊥)
11		mdandyv7.1	. . 3 ⊢ (𝜑 ↔ ⊥)
12	10, 11	bothfbothsame 44395	. 2 ⊢ (𝜂 ↔ 𝜑)
13	9, 12	pm3.2i 471	1 ⊢ ((((𝜒 ↔ 𝜓) ∧ (𝜃 ↔ 𝜓)) ∧ (𝜏 ↔ 𝜓)) ∧ (𝜂 ↔ 𝜑))

Syntax hints:   ↔ wb 205   ∧ wa 396  ⊤wtru 1540  ⊥wfal 1551

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 397

This theorem is referenced by: (None)