Theorem rcp-NDBIEF0 240

Description: '↔' Elimination by Forward Implication Introduction. †

Hypothesis

Ref	Expression
rcp-NDBIEF0.1	⊢ (𝜑 ↔ 𝜓)

Assertion

Ref	Expression
rcp-NDBIEF0	⊢ (𝜑 → 𝜓)

Proof of Theorem rcp-NDBIEF0

Step	Hyp	Ref	Expression
1		rcp-NDBIEF0.1	. . . 4 ⊢ (𝜑 ↔ 𝜓)
2	1	ndtruei-P3.17 182	. . 3 ⊢ (⊤ → (𝜑 ↔ 𝜓))
3	2	ndbief-P3.14 179	. 2 ⊢ (⊤ → (𝜑 → 𝜓))
4	3	ndtruee-P3.18 183	1 ⊢ (𝜑 → 𝜓)

Syntax hints:   → wff-imp 10   ↔ 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-true-D2.4 155

This theorem is referenced by:  idandtruer-P4.19b  439  idorfalser-P4.20b  441  oroverim-P4.28-L1  465  oroverbiint-P4.28d  471  imoverand-P4.29a  472  imoveror-P4.29-L1  473  rimoverand-P4.31-L1  480  rimoveror-P4.31b  482  peirce2exclmid-P4.41b  513  suballinf-P5  594  subexinf-P5  608  specw-P5  661  lemma-L5.04a  667  exipsub-P6  720  nfrall2-P6  743  nfrex2-P6  744  nfrgencl-L6  811  exgennfrcl-L6  814  nfrimd-P6  815  nfrexgen-P7  931  lemma-L7.02a  944  dfnfreeint-P7  969  dfpsubv-P7  977  cbvall-P7-L1  1060  nfrnegconv-P8  1110