Theorem sbimdOLD 2518

Description: Obsolete version of sbimd as of 9-Jul-2023. (Contributed by Wolf Lammen, 24-Nov-2022.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
sbimdOLD.1	⊢ Ⅎ𝑥𝜑
sbimdOLD.2	⊢ (𝜑 → (𝜓 → 𝜒))

Assertion

Ref	Expression
sbimdOLD	⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜒))

Proof of Theorem sbimdOLD

Step	Hyp	Ref	Expression
1		sbimdOLD.2	. . . 4 ⊢ (𝜑 → (𝜓 → 𝜒))
2	1	imim2d 57	. . 3 ⊢ (𝜑 → ((𝑥 = 𝑦 → 𝜓) → (𝑥 = 𝑦 → 𝜒)))
3		sbimdOLD.1	. . . 4 ⊢ Ⅎ𝑥𝜑
4	1	anim2d 613	. . . 4 ⊢ (𝜑 → ((𝑥 = 𝑦 ∧ 𝜓) → (𝑥 = 𝑦 ∧ 𝜒)))
5	3, 4	eximd 2216	. . 3 ⊢ (𝜑 → (∃𝑥(𝑥 = 𝑦 ∧ 𝜓) → ∃𝑥(𝑥 = 𝑦 ∧ 𝜒)))
6	2, 5	anim12d 610	. 2 ⊢ (𝜑 → (((𝑥 = 𝑦 → 𝜓) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜓)) → ((𝑥 = 𝑦 → 𝜒) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜒))))
7		dfsb1 2510	. 2 ⊢ ([𝑦 / 𝑥]𝜓 ↔ ((𝑥 = 𝑦 → 𝜓) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜓)))
8		dfsb1 2510	. 2 ⊢ ([𝑦 / 𝑥]𝜒 ↔ ((𝑥 = 𝑦 → 𝜒) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜒)))
9	6, 7, 8	3imtr4g 298	1 ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜒))

Syntax hints:   → wi 4   ∧ wa 398  ∃wex 1780  Ⅎwnf 1784  [wsb 2069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2145  ax-12 2177  ax-13 2390

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1781  df-nf 1785  df-sb 2070

This theorem is referenced by: (None)