Theorem spcimgf 3550

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 3547	. 2 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → 𝜓)))
4		spcimgf.3	. 2 ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓))
5	3, 4	mpg 1794	1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → 𝜓))

Syntax hints:   → wi 4  ∀wal 1535   = wceq 1537  Ⅎwnf 1780   ∈ wcel 2106  Ⅎwnfc 2888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1777  df-nf 1781  df-cleq 2727  df-clel 2814  df-nfc 2890

This theorem is referenced by:  spcimegf  3551  iooelexlt  37345