Theorem mdandyvr15 41147

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
mdandyvr15.1	⊢ (𝜑 ↔ 𝜁)
mdandyvr15.2	⊢ (𝜓 ↔ 𝜎)
mdandyvr15.3	⊢ (𝜒 ↔ 𝜓)
mdandyvr15.4	⊢ (𝜃 ↔ 𝜓)
mdandyvr15.5	⊢ (𝜏 ↔ 𝜓)
mdandyvr15.6	⊢ (𝜂 ↔ 𝜓)

Assertion

Ref	Expression
mdandyvr15	⊢ ((((𝜒 ↔ 𝜎) ∧ (𝜃 ↔ 𝜎)) ∧ (𝜏 ↔ 𝜎)) ∧ (𝜂 ↔ 𝜎))

Proof of Theorem mdandyvr15

Step	Hyp	Ref	Expression
1		mdandyvr15.2	. 2 ⊢ (𝜓 ↔ 𝜎)
2		mdandyvr15.1	. 2 ⊢ (𝜑 ↔ 𝜁)
3		mdandyvr15.3	. 2 ⊢ (𝜒 ↔ 𝜓)
4		mdandyvr15.4	. 2 ⊢ (𝜃 ↔ 𝜓)
5		mdandyvr15.5	. 2 ⊢ (𝜏 ↔ 𝜓)
6		mdandyvr15.6	. 2 ⊢ (𝜂 ↔ 𝜓)
7	1, 2, 3, 4, 5, 6	mdandyvr0 41132	1 ⊢ ((((𝜒 ↔ 𝜎) ∧ (𝜃 ↔ 𝜎)) ∧ (𝜏 ↔ 𝜎)) ∧ (𝜂 ↔ 𝜎))

Syntax hints:   ↔ wb 196   ∧ wa 384

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 197  df-an 386

This theorem is referenced by: (None)