Theorem astbstanbst 44291

Description: Given a is equivalent to T., also given that b is equivalent to T, there exists a proof for a and b is equivalent to T. (Contributed by Jarvin Udandy, 29-Aug-2016.)

Hypotheses

Ref	Expression
astbstanbst.1	⊢ (𝜑 ↔ ⊤)
astbstanbst.2	⊢ (𝜓 ↔ ⊤)

Assertion

Ref	Expression
astbstanbst	⊢ ((𝜑 ∧ 𝜓) ↔ ⊤)

Proof of Theorem astbstanbst

Step	Hyp	Ref	Expression
1		astbstanbst.1	. . . 4 ⊢ (𝜑 ↔ ⊤)
2	1	aistia 44279	. . 3 ⊢ 𝜑
3		astbstanbst.2	. . . 4 ⊢ (𝜓 ↔ ⊤)
4	3	aistia 44279	. . 3 ⊢ 𝜓
5	2, 4	pm3.2i 470	. 2 ⊢ (𝜑 ∧ 𝜓)
6	5	bitru 1548	1 ⊢ ((𝜑 ∧ 𝜓) ↔ ⊤)

Syntax hints:   ↔ wb 205   ∧ wa 395  ⊤wtru 1540

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  df-tru 1542

This theorem is referenced by:  dandysum2p2e4  44380