Theorem impoe-P4.4a 377

Description: Variation of Principle of Explosion Using Implication. †

Hypothesis

Ref	Expression
impoe-P4.4a.1	⊢ (𝛾 → ¬ 𝜑)

Assertion

Ref	Expression
impoe-P4.4a	⊢ (𝛾 → (𝜑 → 𝜓))

Proof of Theorem impoe-P4.4a

Step	Hyp	Ref	Expression
1		rcp-NDASM2of2 194	. . 3 ⊢ ((𝛾 ∧ 𝜑) → 𝜑)
2		impoe-P4.4a.1	. . . 4 ⊢ (𝛾 → ¬ 𝜑)
3	2	rcp-NDIMP1add1 208	. . 3 ⊢ ((𝛾 ∧ 𝜑) → ¬ 𝜑)
4	1, 3	ndnege-P3.4 169	. 2 ⊢ ((𝛾 ∧ 𝜑) → 𝜓)
5	4	rcp-NDIMI2 224	1 ⊢ (𝛾 → (𝜑 → 𝜓))

Syntax hints:  ¬ wff-neg 9   → wff-imp 10   ∧ wff-and 132

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

This theorem is referenced by:  impoe-P4.4a.RC  378  impoe-P4.4a.CL  379  falseie-P4.22b  445  imasor-P4.32-L2  486  biasandorint-P4.34b  492  peirce-P4.40  511  exclmid2peirce-P4.41a  512  allnegex-P7-L1  956