Theorem bimpf-P4 531

Description: Modus Ponens with '↔' (forward). †

Hypotheses

Ref	Expression
bimpf-P4.1	⊢ (𝛾 → 𝜑)
bimpf-P4.2	⊢ (𝛾 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
bimpf-P4	⊢ (𝛾 → 𝜓)

Proof of Theorem bimpf-P4

Step	Hyp	Ref	Expression
1		bimpf-P4.1	. 2 ⊢ (𝛾 → 𝜑)
2		bimpf-P4.2	. . 3 ⊢ (𝛾 → (𝜑 ↔ 𝜓))
3	2	ndbief-P3.14 179	. 2 ⊢ (𝛾 → (𝜑 → 𝜓))
4	1, 3	ndime-P3.6 171	1 ⊢ (𝛾 → 𝜓)

Syntax hints:   → wff-imp 10   ↔ wff-bi 104

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

This theorem is referenced by:  bimpf-P4.RC  532  andcomm2-P4  564  orcomm2-P4  566  alloverimex-P5  601  nfrsucc-P6  780  nfradd-P6  781  nfrmult-P6  782  psubsuccv-P6-L1  805  psubaddv-P6-L1  807  psubmultv-P6-L1  809  ndsubaddd-P7  858  ndsubmultd-P7  859  eqsym-P7  936  gennfrcl-P7  963  eqtrns-P7  987  nfrsucc-P8  1119  nfradd-P8  1120  nfrmult-P8  1121