Theorem sbcim2g 45062

Description: Distribution of class substitution over a left-nested implication. Similar to sbcimg 3787. sbcim2g 45062 is sbcim2gVD 45398 without virtual deductions and was automatically derived from sbcim2gVD 45398 using the tools program translate..without..overwriting.cmd and Metamath's minimize command. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
sbcim2g	⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒)) ↔ ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒))))

Proof of Theorem sbcim2g

Step	Hyp	Ref	Expression
1		sbcimg 3787	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒)) ↔ ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥](𝜓 → 𝜒))))
2	1	biimpd 231	. . 3 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒)) → ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥](𝜓 → 𝜒))))
3		sbcimg 3787	. . 3 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝜓 → 𝜒) ↔ ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)))
4		imbi2 350	. . . 4 ⊢ (([𝐴 / 𝑥](𝜓 → 𝜒) ↔ ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)) → (([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥](𝜓 → 𝜒)) ↔ ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒))))
5	4	biimpcd 251	. . 3 ⊢ (([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥](𝜓 → 𝜒)) → (([𝐴 / 𝑥](𝜓 → 𝜒) ↔ ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)) → ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒))))
6	2, 3, 5	syl6ci 71	. 2 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒)) → ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒))))
7		idd 24	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)) → ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒))))
8		biimpr 222	. . . 4 ⊢ (([𝐴 / 𝑥](𝜓 → 𝜒) ↔ ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)) → (([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒) → [𝐴 / 𝑥](𝜓 → 𝜒)))
9	3, 7, 8	ee13 45028	. . 3 ⊢ (𝐴 ∈ 𝑉 → (([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)) → ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥](𝜓 → 𝜒))))
10	9, 1	sylibrd 261	. 2 ⊢ (𝐴 ∈ 𝑉 → (([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)) → [𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒))))
11	6, 10	impbid 214	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒)) ↔ ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒))))

Syntax hints:   → wi 4   ↔ wb 208   ∈ wcel 2136  [wsbc 3739

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-10 2169  ax-12 2206  ax-ext 2728

This theorem depends on definitions:  df-bi 209  df-an 399  df-tru 1557  df-ex 1794  df-nf 1798  df-sb 2085  df-clab 2735  df-cleq 2748  df-clel 2831  df-sbc 3740

This theorem is referenced by:  trsbc  45064  trsbcVD  45400