Theorem thmeqtrue-P4.21a 442

Description: Any Theorem is Logically Equivalent to '⊤'. †

Hypothesis

Ref	Expression
thmeqtrue-P4.21a.1	⊢ 𝜑

Assertion

Ref	Expression
thmeqtrue-P4.21a	⊢ (𝜑 ↔ ⊤)

Proof of Theorem thmeqtrue-P4.21a

Step	Hyp	Ref	Expression
1		true-P3.44 352	. . 3 ⊢ ⊤
2	1	rcp-NDIMP0addall 207	. 2 ⊢ (𝜑 → ⊤)
3		thmeqtrue-P4.21a.1	. . 3 ⊢ 𝜑
4	3	ndtruei-P3.17 182	. 2 ⊢ (⊤ → 𝜑)
5	2, 4	rcp-NDBII0 239	1 ⊢ (𝜑 ↔ ⊤)

Syntax hints:   ↔ wff-bi 104  ⊤wff-true 153

This theorem was proved from axioms:  ax-L1 11  ax-L2 12  ax-L3 13  ax-MP 14

This theorem depends on definitions:  df-bi-D2.1 107  df-and-D2.2 133  df-true-D2.4 155

This theorem is referenced by:  idandthml-P4.23a  446  idandthmr-P4.23b  447  truthtblnegf-P4.35b  494  truthtblfimt-P4.36c  497  truthtblfimf-P4.36d  498  truthtbltbit-P4.39a  507  truthtblfbif-P4.39d  510  solvesub-P6a  704  lemma-L6.02a  726  psubnfr-P6  784  nfrthm-P7  926