Theorem spsbcdi 39030

Description: A lemma for eliminating a universal quantifier, in inference form. (Contributed by Giovanni Mascellani, 30-May-2019.)

Hypotheses

Ref	Expression
spsbcdi.1	⊢ 𝐴 ∈ V
spsbcdi.2	⊢ (𝜑 → ∀𝑥𝜒)
spsbcdi.3	⊢ ([𝐴 / 𝑥]𝜒 ↔ 𝜓)

Assertion

Ref	Expression
spsbcdi	⊢ (𝜑 → 𝜓)

Proof of Theorem spsbcdi

Step	Hyp	Ref	Expression
1		spsbcdi.1	. . . 4 ⊢ 𝐴 ∈ V
2	1	a1i 11	. . 3 ⊢ (𝜑 → 𝐴 ∈ V)
3		spsbcdi.2	. . 3 ⊢ (𝜑 → ∀𝑥𝜒)
4	2, 3	spsbcd 3753	. 2 ⊢ (𝜑 → [𝐴 / 𝑥]𝜒)
5		spsbcdi.3	. 2 ⊢ ([𝐴 / 𝑥]𝜒 ↔ 𝜓)
6	4, 5	sylib 221	1 ⊢ (𝜑 → 𝜓)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1568   ∈ wcel 2145  Vcvv 3451  [wsbc 3739

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-sbc 3740

This theorem is used by: (None)