Theorem exe-P7r.RC 1002

Description: Inference Form of exe-P7r 998. †

Hypotheses

Ref	Expression
exe-P7r.RC.1	⊢ Ⅎ𝑥𝜓
exe-P7r.RC.2	⊢ (𝜑 → 𝜓)
exe-P7r.RC.3	⊢ ∃𝑥𝜑

Assertion

Ref	Expression
exe-P7r.RC	⊢ 𝜓

Proof of Theorem exe-P7r.RC

Step	Hyp	Ref	Expression
1		ndnfrv-P7.1 826	. . 3 ⊢ Ⅎ𝑥⊤
2		exe-P7r.RC.1	. . 3 ⊢ Ⅎ𝑥𝜓
3		exe-P7r.RC.2	. . . 4 ⊢ (𝜑 → 𝜓)
4	3	ndtruei-P3.17 182	. . 3 ⊢ (⊤ → (𝜑 → 𝜓))
5		exe-P7r.RC.3	. . . 4 ⊢ ∃𝑥𝜑
6	5	ndtruei-P3.17 182	. . 3 ⊢ (⊤ → ∃𝑥𝜑)
7	1, 2, 4, 6	exe-P7 955	. 2 ⊢ (⊤ → 𝜓)
8	7	ndtruee-P3.18 183	1 ⊢ 𝜓

Syntax hints:   → wff-imp 10  ⊤wff-true 153  ∃wff-exists 595  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  ax-L5 17  ax-L6 18  ax-L7 19  ax-L10 27  ax-L11 28  ax-L12 29

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  df-psub-D6.2 716

This theorem is referenced by:  exe-P7r.RC.VR  1003