Theorem sn-axprlem3 39158

Description: axprlem3 5326 using only Tarski's FOL axiom schemes and ax-rep 5190. (Contributed by SN, 22-Sep-2023.)

Assertion

Ref	Expression
sn-axprlem3	⊢ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑥 if-(𝜑, 𝑧 = 𝑎, 𝑧 = 𝑏))

Distinct variable groups:   𝑥,𝑤,𝑦,𝑧   𝜑,𝑦,𝑧   𝑦,𝑎,𝑧   𝑦,𝑏,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑤,𝑎,𝑏)

Proof of Theorem sn-axprlem3

Step	Hyp	Ref	Expression
1		axrep6 5197	. 2 ⊢ (∀𝑤∃*𝑧if-(𝜑, 𝑧 = 𝑎, 𝑧 = 𝑏) → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑥 if-(𝜑, 𝑧 = 𝑎, 𝑧 = 𝑏)))
2		ax6evr 2022	. . . . 5 ⊢ ∃𝑦 𝑎 = 𝑦
3		ifptru 1068	. . . . . . . . . 10 ⊢ (𝜑 → (if-(𝜑, 𝑧 = 𝑎, 𝑧 = 𝑏) ↔ 𝑧 = 𝑎))
4	3	biimpd 231	. . . . . . . . 9 ⊢ (𝜑 → (if-(𝜑, 𝑧 = 𝑎, 𝑧 = 𝑏) → 𝑧 = 𝑎))
5		equtrr 2029	. . . . . . . . 9 ⊢ (𝑎 = 𝑦 → (𝑧 = 𝑎 → 𝑧 = 𝑦))
6	4, 5	sylan9r 511	. . . . . . . 8 ⊢ ((𝑎 = 𝑦 ∧ 𝜑) → (if-(𝜑, 𝑧 = 𝑎, 𝑧 = 𝑏) → 𝑧 = 𝑦))
7	6	alrimiv 1928	. . . . . . 7 ⊢ ((𝑎 = 𝑦 ∧ 𝜑) → ∀𝑧(if-(𝜑, 𝑧 = 𝑎, 𝑧 = 𝑏) → 𝑧 = 𝑦))
8	7	expcom 416	. . . . . 6 ⊢ (𝜑 → (𝑎 = 𝑦 → ∀𝑧(if-(𝜑, 𝑧 = 𝑎, 𝑧 = 𝑏) → 𝑧 = 𝑦)))
9	8	eximdv 1918	. . . . 5 ⊢ (𝜑 → (∃𝑦 𝑎 = 𝑦 → ∃𝑦∀𝑧(if-(𝜑, 𝑧 = 𝑎, 𝑧 = 𝑏) → 𝑧 = 𝑦)))
10	2, 9	mpi 20	. . . 4 ⊢ (𝜑 → ∃𝑦∀𝑧(if-(𝜑, 𝑧 = 𝑎, 𝑧 = 𝑏) → 𝑧 = 𝑦))
11		ax6evr 2022	. . . . 5 ⊢ ∃𝑦 𝑏 = 𝑦
12		ifpfal 1069	. . . . . . . . . 10 ⊢ (¬ 𝜑 → (if-(𝜑, 𝑧 = 𝑎, 𝑧 = 𝑏) ↔ 𝑧 = 𝑏))
13	12	biimpd 231	. . . . . . . . 9 ⊢ (¬ 𝜑 → (if-(𝜑, 𝑧 = 𝑎, 𝑧 = 𝑏) → 𝑧 = 𝑏))
14		equtrr 2029	. . . . . . . . 9 ⊢ (𝑏 = 𝑦 → (𝑧 = 𝑏 → 𝑧 = 𝑦))
15	13, 14	sylan9r 511	. . . . . . . 8 ⊢ ((𝑏 = 𝑦 ∧ ¬ 𝜑) → (if-(𝜑, 𝑧 = 𝑎, 𝑧 = 𝑏) → 𝑧 = 𝑦))
16	15	alrimiv 1928	. . . . . . 7 ⊢ ((𝑏 = 𝑦 ∧ ¬ 𝜑) → ∀𝑧(if-(𝜑, 𝑧 = 𝑎, 𝑧 = 𝑏) → 𝑧 = 𝑦))
17	16	expcom 416	. . . . . 6 ⊢ (¬ 𝜑 → (𝑏 = 𝑦 → ∀𝑧(if-(𝜑, 𝑧 = 𝑎, 𝑧 = 𝑏) → 𝑧 = 𝑦)))
18	17	eximdv 1918	. . . . 5 ⊢ (¬ 𝜑 → (∃𝑦 𝑏 = 𝑦 → ∃𝑦∀𝑧(if-(𝜑, 𝑧 = 𝑎, 𝑧 = 𝑏) → 𝑧 = 𝑦)))
19	11, 18	mpi 20	. . . 4 ⊢ (¬ 𝜑 → ∃𝑦∀𝑧(if-(𝜑, 𝑧 = 𝑎, 𝑧 = 𝑏) → 𝑧 = 𝑦))
20	10, 19	pm2.61i 184	. . 3 ⊢ ∃𝑦∀𝑧(if-(𝜑, 𝑧 = 𝑎, 𝑧 = 𝑏) → 𝑧 = 𝑦)
21		df-mo 2622	. . 3 ⊢ (∃*𝑧if-(𝜑, 𝑧 = 𝑎, 𝑧 = 𝑏) ↔ ∃𝑦∀𝑧(if-(𝜑, 𝑧 = 𝑎, 𝑧 = 𝑏) → 𝑧 = 𝑦))
22	20, 21	mpbir 233	. 2 ⊢ ∃*𝑧if-(𝜑, 𝑧 = 𝑎, 𝑧 = 𝑏)
23	1, 22	mpg 1798	1 ⊢ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑥 if-(𝜑, 𝑧 = 𝑎, 𝑧 = 𝑏))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398  if-wif 1057  ∀wal 1535  ∃wex 1780  ∃*wmo 2620  ∃wrex 3139

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-rep 5190

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ifp 1058  df-ex 1781  df-mo 2622  df-rex 3144

This theorem is referenced by: (None)