Theorem mdandyv4 44448

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
mdandyv4.1	⊢ (𝜑 ↔ ⊥)
mdandyv4.2	⊢ (𝜓 ↔ ⊤)
mdandyv4.3	⊢ (𝜒 ↔ ⊥)
mdandyv4.4	⊢ (𝜃 ↔ ⊥)
mdandyv4.5	⊢ (𝜏 ↔ ⊤)
mdandyv4.6	⊢ (𝜂 ↔ ⊥)

Assertion

Ref	Expression
mdandyv4	⊢ ((((𝜒 ↔ 𝜑) ∧ (𝜃 ↔ 𝜑)) ∧ (𝜏 ↔ 𝜓)) ∧ (𝜂 ↔ 𝜑))

Proof of Theorem mdandyv4

Step	Hyp	Ref	Expression
1		mdandyv4.3	. . . . 5 ⊢ (𝜒 ↔ ⊥)
2		mdandyv4.1	. . . . 5 ⊢ (𝜑 ↔ ⊥)
3	1, 2	bothfbothsame 44395	. . . 4 ⊢ (𝜒 ↔ 𝜑)
4		mdandyv4.4	. . . . 5 ⊢ (𝜃 ↔ ⊥)
5	4, 2	bothfbothsame 44395	. . . 4 ⊢ (𝜃 ↔ 𝜑)
6	3, 5	pm3.2i 471	. . 3 ⊢ ((𝜒 ↔ 𝜑) ∧ (𝜃 ↔ 𝜑))
7		mdandyv4.5	. . . 4 ⊢ (𝜏 ↔ ⊤)
8		mdandyv4.2	. . . 4 ⊢ (𝜓 ↔ ⊤)
9	7, 8	bothtbothsame 44394	. . 3 ⊢ (𝜏 ↔ 𝜓)
10	6, 9	pm3.2i 471	. 2 ⊢ (((𝜒 ↔ 𝜑) ∧ (𝜃 ↔ 𝜑)) ∧ (𝜏 ↔ 𝜓))
11		mdandyv4.6	. . 3 ⊢ (𝜂 ↔ ⊥)
12	11, 2	bothfbothsame 44395	. 2 ⊢ (𝜂 ↔ 𝜑)
13	10, 12	pm3.2i 471	1 ⊢ ((((𝜒 ↔ 𝜑) ∧ (𝜃 ↔ 𝜑)) ∧ (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)