Theorem suballv-P5 623

Description: Substitution Law for '∀𝑥' (variable restriction). The most general form is suball-P6 753.

Hypothesis

Ref	Expression
suballv-P5.1	⊢ (𝛾 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
suballv-P5	⊢ (𝛾 → (∀𝑥𝜑 ↔ ∀𝑥𝜓))

Distinct variable group:   𝛾,𝑥

Proof of Theorem suballv-P5

Step	Hyp	Ref	Expression
1		suballv-P5.1	. . . 4 ⊢ (𝛾 → (𝜑 ↔ 𝜓))
2	1	ndbief-P3.14 179	. . 3 ⊢ (𝛾 → (𝜑 → 𝜓))
3	2	alloverim-P5.GENV 621	. 2 ⊢ (𝛾 → (∀𝑥𝜑 → ∀𝑥𝜓))
4	1	ndbier-P3.15 180	. . 3 ⊢ (𝛾 → (𝜓 → 𝜑))
5	4	alloverim-P5.GENV 621	. 2 ⊢ (𝛾 → (∀𝑥𝜓 → ∀𝑥𝜑))
6	3, 5	ndbii-P3.13 178	1 ⊢ (𝛾 → (∀𝑥𝜑 ↔ ∀𝑥𝜓))

Syntax hints:  ∀wff-forall 8   → wff-imp 10   ↔ wff-bi 104

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

This theorem depends on definitions:  df-bi-D2.1 107  df-and-D2.2 133  df-true-D2.4 155

This theorem is referenced by:  example-E5.02a  664  example-E5.03a  665  example-E5.04a  675  nfrall2w-P6  694  example-E6.01a  706  psubjust-P6  715  dfpsubv-P6  717  psuball2v-P6-L1  795