Theorem spsbcdi 33923

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 3449	. 2 ⊢ (𝜑 → [𝐴 / 𝑥]𝜒)
5		spsbcdi.3	. 2 ⊢ ([𝐴 / 𝑥]𝜒 ↔ 𝜓)
6	4, 5	sylib 208	1 ⊢ (𝜑 → 𝜓)

Syntax hints:   → wi 4   ↔ wb 196  ∀wal 1481   ∈ wcel 1990  Vcvv 3200  [wsbc 3435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-an 386  df-tru 1486  df-ex 1705  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-v 3202  df-sbc 3436

This theorem is referenced by: (None)