Theorem ndexew-P7.RC 887

Description: Inference Form of ndexew-P7 867. †

Hypotheses

Ref	Expression
ndexew-P7.RC.1	⊢ Ⅎ𝑦𝜑
ndexew-P7.RC.2	⊢ Ⅎ𝑦𝜓
ndexew-P7.RC.3	⊢ ([𝑦 / 𝑥]𝜑 → 𝜓)
ndexew-P7.RC.4	⊢ ∃𝑥𝜑

Assertion

Ref	Expression
ndexew-P7.RC	⊢ 𝜓

Distinct variable group:   𝑥,𝑦

Proof of Theorem ndexew-P7.RC

Step	Hyp	Ref	Expression
1		ndexew-P7.RC.1	. . 3 ⊢ Ⅎ𝑦𝜑
2		ndexew-P7.RC.2	. . 3 ⊢ Ⅎ𝑦𝜓
3		ndexew-P7.RC.3	. . . 4 ⊢ ([𝑦 / 𝑥]𝜑 → 𝜓)
4	3	ndtruei-P3.17 182	. . 3 ⊢ (⊤ → ([𝑦 / 𝑥]𝜑 → 𝜓))
5		ndexew-P7.RC.4	. . . 4 ⊢ ∃𝑥𝜑
6	5	ndtruei-P3.17 182	. . 3 ⊢ (⊤ → ∃𝑥𝜑)
7	1, 2, 4, 6	ndexew-P7.VR1of3 868	. 2 ⊢ (⊤ → 𝜓)
8	7	ndtruee-P3.18 183	1 ⊢ 𝜓

Syntax hints:  term-obj 1   → wff-imp 10  ⊤wff-true 153  ∃wff-exists 595  Ⅎwff-nfree 681  [wff-psub 714

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:  ndexew-P7.RC.VR1of2  888  ndexew-P7.RC.VR2of2  889  ndexew-P7.RC.VR12of2  890