Theorem mdandyv9 44340

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
mdandyv9.1	⊢ (𝜑 ↔ ⊥)
mdandyv9.2	⊢ (𝜓 ↔ ⊤)
mdandyv9.3	⊢ (𝜒 ↔ ⊤)
mdandyv9.4	⊢ (𝜃 ↔ ⊥)
mdandyv9.5	⊢ (𝜏 ↔ ⊥)
mdandyv9.6	⊢ (𝜂 ↔ ⊤)

Assertion

Ref	Expression
mdandyv9	⊢ ((((𝜒 ↔ 𝜓) ∧ (𝜃 ↔ 𝜑)) ∧ (𝜏 ↔ 𝜑)) ∧ (𝜂 ↔ 𝜓))

Proof of Theorem mdandyv9

Step	Hyp	Ref	Expression
1		mdandyv9.3	. . . . 5 ⊢ (𝜒 ↔ ⊤)
2		mdandyv9.2	. . . . 5 ⊢ (𝜓 ↔ ⊤)
3	1, 2	bothtbothsame 44281	. . . 4 ⊢ (𝜒 ↔ 𝜓)
4		mdandyv9.4	. . . . 5 ⊢ (𝜃 ↔ ⊥)
5		mdandyv9.1	. . . . 5 ⊢ (𝜑 ↔ ⊥)
6	4, 5	bothfbothsame 44282	. . . 4 ⊢ (𝜃 ↔ 𝜑)
7	3, 6	pm3.2i 470	. . 3 ⊢ ((𝜒 ↔ 𝜓) ∧ (𝜃 ↔ 𝜑))
8		mdandyv9.5	. . . 4 ⊢ (𝜏 ↔ ⊥)
9	8, 5	bothfbothsame 44282	. . 3 ⊢ (𝜏 ↔ 𝜑)
10	7, 9	pm3.2i 470	. 2 ⊢ (((𝜒 ↔ 𝜓) ∧ (𝜃 ↔ 𝜑)) ∧ (𝜏 ↔ 𝜑))
11		mdandyv9.6	. . 3 ⊢ (𝜂 ↔ ⊤)
12	11, 2	bothtbothsame 44281	. 2 ⊢ (𝜂 ↔ 𝜓)
13	10, 12	pm3.2i 470	1 ⊢ ((((𝜒 ↔ 𝜓) ∧ (𝜃 ↔ 𝜑)) ∧ (𝜏 ↔ 𝜑)) ∧ (𝜂 ↔ 𝜓))

Syntax hints:   ↔ wb 205   ∧ wa 395  ⊤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 396

This theorem is referenced by: (None)