Theorem sbimdvOLD 2516

Description: Obsolete version of sbimdv 2083 as of 6-Jul-2023. Deduction substituting both sides of an implication, with 𝜑 and 𝑥 disjoint. See also sbimd 2245. (Contributed by Wolf Lammen, 6-May-2023.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
sbimdvOLD.2	⊢ (𝜑 → (𝜓 → 𝜒))

Assertion

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

Distinct variable group:   𝜑,𝑥

Allowed substitution hints:   𝜑(𝑦)   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦)

Proof of Theorem sbimdvOLD

Step	Hyp	Ref	Expression
1		sbimdvOLD.2	. . . 4 ⊢ (𝜑 → (𝜓 → 𝜒))
2	1	imim2d 57	. . 3 ⊢ (𝜑 → ((𝑥 = 𝑦 → 𝜓) → (𝑥 = 𝑦 → 𝜒)))
3	1	anim2d 613	. . . 4 ⊢ (𝜑 → ((𝑥 = 𝑦 ∧ 𝜓) → (𝑥 = 𝑦 ∧ 𝜒)))
4	3	eximdv 1918	. . 3 ⊢ (𝜑 → (∃𝑥(𝑥 = 𝑦 ∧ 𝜓) → ∃𝑥(𝑥 = 𝑦 ∧ 𝜒)))
5	2, 4	anim12d 610	. 2 ⊢ (𝜑 → (((𝑥 = 𝑦 → 𝜓) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜓)) → ((𝑥 = 𝑦 → 𝜒) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜒))))
6		dfsb1 2510	. 2 ⊢ ([𝑦 / 𝑥]𝜓 ↔ ((𝑥 = 𝑦 → 𝜓) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜓)))
7		dfsb1 2510	. 2 ⊢ ([𝑦 / 𝑥]𝜒 ↔ ((𝑥 = 𝑦 → 𝜒) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜒)))
8	5, 6, 7	3imtr4g 298	1 ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜒))

Syntax hints:   → wi 4   ∧ wa 398  ∃wex 1780  [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)