Theorem spsbcdi 38109

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 3775	. 2 ⊢ (𝜑 → [𝐴 / 𝑥]𝜒)
5		spsbcdi.3	. 2 ⊢ ([𝐴 / 𝑥]𝜒 ↔ 𝜓)
6	4, 5	sylib 218	1 ⊢ (𝜑 → 𝜓)

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1538   ∈ wcel 2109  Vcvv 3455  [wsbc 3761

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-sbc 3762

This theorem is referenced by: (None)