Theorem falseprofelimr-P4.7b 395

Description: Process of Elimination Utilizing '⊥' (right). †

Hypotheses

Ref	Expression
falseprofelimr-P4.7b.1	⊢ (𝛾 → (𝜑 ∨ 𝜓))
falseprofelimr-P4.7b.2	⊢ (𝛾 → (𝜓 → ⊥))

Assertion

Ref	Expression
falseprofelimr-P4.7b	⊢ (𝛾 → 𝜑)

Proof of Theorem falseprofelimr-P4.7b

Step	Hyp	Ref	Expression
1		falseprofelimr-P4.7b.1	. 2 ⊢ (𝛾 → (𝜑 ∨ 𝜓))
2		rcp-NDASM2of2 194	. . . 4 ⊢ ((𝛾 ∧ 𝜓) → 𝜓)
3		falseprofelimr-P4.7b.2	. . . . 5 ⊢ (𝛾 → (𝜓 → ⊥))
4	3	rcp-NDIMP1add1 208	. . . 4 ⊢ ((𝛾 ∧ 𝜓) → (𝜓 → ⊥))
5	2, 4	ndime-P3.6 171	. . 3 ⊢ ((𝛾 ∧ 𝜓) → ⊥)
6	5	falseimpoe-P4.4c 383	. . 3 ⊢ ((𝛾 ∧ 𝜓) → ¬ ⊥)
7	5, 6	rcp-NDNEGI2 219	. 2 ⊢ (𝛾 → ¬ 𝜓)
8	1, 7	profelimr-P4.5b 387	1 ⊢ (𝛾 → 𝜑)

Syntax hints:  ¬ wff-neg 9   → wff-imp 10   ∧ wff-and 132   ∨ wff-or 144  ⊥wff-false 157

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-or-D2.3 145  df-true-D2.4 155  df-false-D2.5 158

This theorem is referenced by:  falseprofelimr-P4.7b.RC  396