Theorem 19.21 1562

Description: Theorem 19.21 of [Margaris] p. 90. The hypothesis can be thought of as "𝑥 is not free in 𝜑." (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)

Hypothesis

Ref	Expression
19.21.1	⊢ Ⅎ𝑥𝜑

Assertion

Ref	Expression
19.21	⊢ (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓))

Proof of Theorem 19.21

Step	Hyp	Ref	Expression
1		19.21.1	. 2 ⊢ Ⅎ𝑥𝜑
2		19.21t 1561	. 2 ⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓)))
3	1, 2	ax-mp 5	1 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓))

Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1329  Ⅎwnf 1436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-4 1487  ax-ial 1514  ax-i5r 1515

This theorem depends on definitions:  df-bi 116  df-nf 1437

This theorem is referenced by:  stdpc5  1563  19.21-2  1645  19.32dc  1657  cbv1  1722  eu2  2041  mo3h  2050  moanim  2071  r2alf  2450