Theorem sbcimi 38570

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 3790	. . 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 352	. 2 ⊢ (([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓) ↔ (𝜒 → 𝜂))
7	3, 6	bitri 277	1 ⊢ ([𝐴 / 𝑥](𝜑 → 𝜓) ↔ (𝜒 → 𝜂))

Syntax hints:   → wi 4   ↔ wb 208   ∈ wcel 2141  Vcvv 3453  [wsbc 3742

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-12 2211  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1562  df-ex 1799  df-nf 1803  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-sbc 3743

This theorem is referenced by: (None)