Theorem sbcimi 33912

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 3477	. . 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 340	. 2 ⊢ (([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓) ↔ (𝜒 → 𝜂))
7	3, 6	bitri 264	1 ⊢ ([𝐴 / 𝑥](𝜑 → 𝜓) ↔ (𝜒 → 𝜂))

Syntax hints:   → wi 4   ↔ wb 196   ∈ 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-10 2019  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-v 3202  df-sbc 3436

This theorem is referenced by: (None)