Theorem spcimgf 3508

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 3505	. 2 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → 𝜓)))
4		spcimgf.3	. 2 ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓))
5	3, 4	mpg 1799	1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → 𝜓))

Syntax hints:   → wi 4  ∀wal 1540   = wceq 1542  Ⅎwnf 1785   ∈ wcel 2114  Ⅎwnfc 2884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ex 1782  df-nf 1786  df-cleq 2729  df-clel 2812  df-nfc 2886

This theorem is referenced by:  spcimegf  3509  iooelexlt  37569