Theorem ndnfrand-P7.4.RC 877

Description: Inference Form of ndnfrand-P7.4 829. †

Hypotheses

Ref	Expression
ndnfrand-P7.4.RC.1	⊢ Ⅎ𝑥𝜑
ndnfrand-P7.4.RC.2	⊢ Ⅎ𝑥𝜓

Assertion

Ref	Expression
ndnfrand-P7.4.RC	⊢ Ⅎ𝑥(𝜑 ∧ 𝜓)

Proof of Theorem ndnfrand-P7.4.RC

Step	Hyp	Ref	Expression
1		ndnfrand-P7.4.RC.1	. . . 4 ⊢ Ⅎ𝑥𝜑
2	1	ndtruei-P3.17 182	. . 3 ⊢ (⊤ → Ⅎ𝑥𝜑)
3		ndnfrand-P7.4.RC.2	. . . 4 ⊢ Ⅎ𝑥𝜓
4	3	ndtruei-P3.17 182	. . 3 ⊢ (⊤ → Ⅎ𝑥𝜓)
5	2, 4	ndnfrand-P7.4 829	. 2 ⊢ (⊤ → Ⅎ𝑥(𝜑 ∧ 𝜓))
6	5	ndtruee-P3.18 183	1 ⊢ Ⅎ𝑥(𝜑 ∧ 𝜓)

Syntax hints:   ∧ wff-and 132  ⊤wff-true 153  Ⅎwff-nfree 681

This theorem was proved from axioms:  ax-L1 11  ax-L2 12  ax-L3 13  ax-MP 14  ax-GEN 15  ax-L4 16

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-rcp-AND3 161  df-exists-D5.1 596  df-nfree-D6.1 682

This theorem is referenced by:  allnegex-P7-L1  956  allic-P7  1007  exnegallint-P7  1047  qimeqex-P7-L1  1054  qimeqex-P7-L2  1055  nfradd-P8  1120  nfrmult-P8  1121