Theorem suball-P7 973

Description: Substitution Theorem for '∀𝑥' (non-freeness condition). †

Hypotheses

Ref	Expression
suball-P7.1	⊢ Ⅎ𝑥𝛾
suball-P7.2	⊢ (𝛾 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
suball-P7	⊢ (𝛾 → (∀𝑥𝜑 ↔ ∀𝑥𝜓))

Proof of Theorem suball-P7

Step	Hyp	Ref	Expression
1		suball-P7.1	. . 3 ⊢ Ⅎ𝑥𝛾
2		suball-P7.2	. . . 4 ⊢ (𝛾 → (𝜑 ↔ 𝜓))
3	2	ndbief-P3.14 179	. . 3 ⊢ (𝛾 → (𝜑 → 𝜓))
4	1, 3	alloverim-P7.GENF 971	. 2 ⊢ (𝛾 → (∀𝑥𝜑 → ∀𝑥𝜓))
5	2	ndbier-P3.15 180	. . 3 ⊢ (𝛾 → (𝜓 → 𝜑))
6	1, 5	alloverim-P7.GENF 971	. 2 ⊢ (𝛾 → (∀𝑥𝜓 → ∀𝑥𝜑))
7	4, 6	ndbii-P3.13 178	1 ⊢ (𝛾 → (∀𝑥𝜑 ↔ ∀𝑥𝜓))

Syntax hints:  ∀wff-forall 8   → wff-imp 10   ↔ wff-bi 104  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:  suball-P7.RC  974  suball-P7r  1039  suball-P7r.VR  1040