Theorem sbim 1887

Description: Implication inside and outside of substitution are equivalent. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 3-Feb-2018.)

Assertion

Ref	Expression
sbim	⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))

Proof of Theorem sbim

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbimv 1832	. . . 4 ⊢ ([𝑧 / 𝑥](𝜑 → 𝜓) ↔ ([𝑧 / 𝑥]𝜑 → [𝑧 / 𝑥]𝜓))
2	1	sbbii 1706	. . 3 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥](𝜑 → 𝜓) ↔ [𝑦 / 𝑧]([𝑧 / 𝑥]𝜑 → [𝑧 / 𝑥]𝜓))
3		sbimv 1832	. . 3 ⊢ ([𝑦 / 𝑧]([𝑧 / 𝑥]𝜑 → [𝑧 / 𝑥]𝜓) ↔ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 → [𝑦 / 𝑧][𝑧 / 𝑥]𝜓))
4	2, 3	bitri 183	. 2 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥](𝜑 → 𝜓) ↔ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 → [𝑦 / 𝑧][𝑧 / 𝑥]𝜓))
5		ax-17 1474	. . 3 ⊢ ((𝜑 → 𝜓) → ∀𝑧(𝜑 → 𝜓))
6	5	sbco2v 1881	. 2 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥](𝜑 → 𝜓) ↔ [𝑦 / 𝑥](𝜑 → 𝜓))
7		ax-17 1474	. . . 4 ⊢ (𝜑 → ∀𝑧𝜑)
8	7	sbco2v 1881	. . 3 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
9		ax-17 1474	. . . 4 ⊢ (𝜓 → ∀𝑧𝜓)
10	9	sbco2v 1881	. . 3 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜓 ↔ [𝑦 / 𝑥]𝜓)
11	8, 10	imbi12i 238	. 2 ⊢ (([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 → [𝑦 / 𝑧][𝑧 / 𝑥]𝜓) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
12	4, 6, 11	3bitr3i 209	1 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))

Syntax hints:   → wi 4   ↔ wb 104  [wsb 1703

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483

This theorem depends on definitions:  df-bi 116  df-nf 1405  df-sb 1704

This theorem is referenced by:  sbrim  1890  sblim  1891  sbbi  1893  moimv  2026  nfraldya  2428  sbcimg  2902  zfregfr  4426  tfi  4434  peano2  4447