Theorem sbcimi 33082

Description: Distribution of class substitution over implication, in inference form. (Contributed by Giovanni Mascellani, 27-May-2019.)

Hypotheses

Ref	Expression
sbcimi.1	⊢ 𝐴 ∈ V
sbcimi.2	⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜒)
sbcimi.3	⊢ ([𝐴 / 𝑥]𝜓 ↔ 𝜂)

Assertion

Ref	Expression
sbcimi	⊢ ([𝐴 / 𝑥](𝜑 → 𝜓) ↔ (𝜒 → 𝜂))

Proof of Theorem sbcimi

Step	Hyp	Ref	Expression
1		sbcimi.1	. . 3 ⊢ 𝐴 ∈ V
2		sbcimg 3444	. . 3 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥](𝜑 → 𝜓) ↔ ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓)))
3	1, 2	ax-mp 5	. 2 ⊢ ([𝐴 / 𝑥](𝜑 → 𝜓) ↔ ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓))
4		sbcimi.2	. . 3 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜒)
5		sbcimi.3	. . 3 ⊢ ([𝐴 / 𝑥]𝜓 ↔ 𝜂)
6	4, 5	imbi12i 339	. 2 ⊢ (([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓) ↔ (𝜒 → 𝜂))
7	3, 6	bitri 263	1 ⊢ ([𝐴 / 𝑥](𝜑 → 𝜓) ↔ (𝜒 → 𝜂))

Syntax hints:   → wi 4   ↔ wb 195   ∈ wcel 1977  Vcvv 3173  [wsbc 3402

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-v 3175  df-sbc 3403

This theorem is referenced by: (None)