Theorem spcimgf 3517

Description: Rule of specialization, using implicit substitution. Compare Theorem 7.3 of [Quine] p. 44. (Contributed by Mario Carneiro, 4-Jan-2017.)

Hypotheses

Ref	Expression
spcimgf.1	⊢ Ⅎ𝑥𝐴
spcimgf.2	⊢ Ⅎ𝑥𝜓
spcimgf.3	⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓))

Assertion

Ref	Expression
spcimgf	⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → 𝜓))

Proof of Theorem spcimgf

Step	Hyp	Ref	Expression
1		spcimgf.2	. . 3 ⊢ Ⅎ𝑥𝜓
2		spcimgf.1	. . 3 ⊢ Ⅎ𝑥𝐴
3	1, 2	spcimgfi1 3515	. 2 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → 𝜓)))
4		spcimgf.3	. 2 ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓))
5	3, 4	mpg 1826	1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → 𝜓))

This proof depends on syntax axioms:   → wi 4  ∀wal 1567   = wceq 1569  Ⅎwnf 1812   ∈ wcel 2142  Ⅎwnfc 2909

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ex 1809  df-nf 1813  df-cleq 2754  df-clel 2837  df-nfc 2911

This theorem is used by:  spcimegf  3518  iooelexlt  38036