Theorem ndexew-P7.VR2of3 869

Description: ndexew-P7 867 with one variable restriction. †

'𝑦' cannot occur in '𝜑'.

Hypotheses

Ref	Expression
ndexew-P7.VR2of3.1	⊢ Ⅎ𝑦𝛾
ndexew-P7.VR2of3.3	⊢ Ⅎ𝑦𝜓
ndexew-P7.VR2of3.4	⊢ (𝛾 → ([𝑦 / 𝑥]𝜑 → 𝜓))
ndexew-P7.VR2of3.5	⊢ (𝛾 → ∃𝑥𝜑)

Assertion

Ref	Expression
ndexew-P7.VR2of3	⊢ (𝛾 → 𝜓)

Distinct variable groups:   𝜑,𝑦   𝑥,𝑦

Proof of Theorem ndexew-P7.VR2of3

Step	Hyp	Ref	Expression
1		ndexew-P7.VR2of3.1	. 2 ⊢ Ⅎ𝑦𝛾
2		ndnfrv-P7.1 826	. 2 ⊢ Ⅎ𝑦𝜑
3		ndexew-P7.VR2of3.3	. 2 ⊢ Ⅎ𝑦𝜓
4		ndexew-P7.VR2of3.4	. 2 ⊢ (𝛾 → ([𝑦 / 𝑥]𝜑 → 𝜓))
5		ndexew-P7.VR2of3.5	. 2 ⊢ (𝛾 → ∃𝑥𝜑)
6	1, 2, 3, 4, 5	ndexew-P7 867	1 ⊢ (𝛾 → 𝜓)

Syntax hints:  term-obj 1   → wff-imp 10  ∃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:  lemma-L7.02a-L1  943