Theorem rmoanimALT 3872

Description: Alternate proof of rmoanim 3871, shorter but requiring ax-10 2144 and ax-11 2160. (Contributed by Alexander van der Vekens, 25-Jun-2017.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
rmoanim.1	⊢ Ⅎ𝑥𝜑

Assertion

Ref	Expression
rmoanimALT	⊢ (∃*𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (𝜑 → ∃*𝑥 ∈ 𝐴 𝜓))

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem rmoanimALT

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		impexp 453	. . . . 5 ⊢ (((𝜑 ∧ 𝜓) → 𝑥 = 𝑦) ↔ (𝜑 → (𝜓 → 𝑥 = 𝑦)))
2	1	ralbii 3164	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦) ↔ ∀𝑥 ∈ 𝐴 (𝜑 → (𝜓 → 𝑥 = 𝑦)))
3		rmoanim.1	. . . . 5 ⊢ Ⅎ𝑥𝜑
4	3	r19.21 3214	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → (𝜓 → 𝑥 = 𝑦)) ↔ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦)))
5	2, 4	bitri 277	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦) ↔ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦)))
6	5	exbii 1847	. 2 ⊢ (∃𝑦∀𝑥 ∈ 𝐴 ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦) ↔ ∃𝑦(𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦)))
7		nfv 1914	. . 3 ⊢ Ⅎ𝑦(𝜑 ∧ 𝜓)
8	7	rmo2 3864	. 2 ⊢ (∃*𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ ∃𝑦∀𝑥 ∈ 𝐴 ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦))
9		nfv 1914	. . . . 5 ⊢ Ⅎ𝑦𝜓
10	9	rmo2 3864	. . . 4 ⊢ (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑦∀𝑥 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦))
11	10	imbi2i 338	. . 3 ⊢ ((𝜑 → ∃*𝑥 ∈ 𝐴 𝜓) ↔ (𝜑 → ∃𝑦∀𝑥 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦)))
12		19.37v 1997	. . 3 ⊢ (∃𝑦(𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦)) ↔ (𝜑 → ∃𝑦∀𝑥 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦)))
13	11, 12	bitr4i 280	. 2 ⊢ ((𝜑 → ∃*𝑥 ∈ 𝐴 𝜓) ↔ ∃𝑦(𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦)))
14	6, 8, 13	3bitr4i 305	1 ⊢ (∃*𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (𝜑 → ∃*𝑥 ∈ 𝐴 𝜓))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  ∃wex 1779  Ⅎwnf 1783  ∀wral 3137  ∃*wrmo 3140

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-10 2144  ax-11 2160  ax-12 2176

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-ex 1780  df-nf 1784  df-mo 2621  df-ral 3142  df-rmo 3145

This theorem is referenced by: (None)