Theorem ndexew-P7 867

Description: ndexe-P7.20 845 with '𝑦' ENF in '𝜓'. †

In the original '∃' Elimination Law, '𝑦' need only be *conditionally* ENF, i.e. '(𝛾 → Ⅎ𝑦𝜓)'.

Hypotheses

Ref	Expression
ndexew-P7.1	⊢ Ⅎ𝑦𝛾
ndexew-P7.2	⊢ Ⅎ𝑦𝜑
ndexew-P7.3	⊢ Ⅎ𝑦𝜓
ndexew-P7.4	⊢ (𝛾 → ([𝑦 / 𝑥]𝜑 → 𝜓))
ndexew-P7.5	⊢ (𝛾 → ∃𝑥𝜑)

Assertion

Ref	Expression
ndexew-P7	⊢ (𝛾 → 𝜓)

Distinct variable group:   𝑥,𝑦

Proof of Theorem ndexew-P7

Step	Hyp	Ref	Expression
1		ndexew-P7.1	. 2 ⊢ Ⅎ𝑦𝛾
2		ndexew-P7.2	. 2 ⊢ Ⅎ𝑦𝜑
3		ndexew-P7.3	. . 3 ⊢ Ⅎ𝑦𝜓
4	3	rcp-NDIMP0addall 207	. 2 ⊢ (𝛾 → Ⅎ𝑦𝜓)
5		ndexew-P7.4	. 2 ⊢ (𝛾 → ([𝑦 / 𝑥]𝜑 → 𝜓))
6		ndexew-P7.5	. 2 ⊢ (𝛾 → ∃𝑥𝜑)
7	1, 2, 4, 5, 6	ndexe-P7.20 845	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:  ndexew-P7.VR1of3  868  ndexew-P7.VR2of3  869  ndexew-P7.VR3of3  870  ndexew-P7.VR12of3  871  ndexew-P7.VR13of3  872  ndexew-P7.VR23of3  873  ndexew-P7.VR123of3  874