Theorem sn-axprlem3 43047

Description: axprlem3 5398 using only Tarski's FOL axiom schemes and ax-rep 5240. (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 5249	. 2 ⊢ (∀𝑤∃*𝑧if-(𝜑, 𝑧 = 𝑎, 𝑧 = 𝑏) → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑥 if-(𝜑, 𝑧 = 𝑎, 𝑧 = 𝑏)))
2		ax6evr 2048	. . . . 5 ⊢ ∃𝑦 𝑎 = 𝑦
3		ifptru 1091	. . . . . . . . . 10 ⊢ (𝜑 → (if-(𝜑, 𝑧 = 𝑎, 𝑧 = 𝑏) ↔ 𝑧 = 𝑎))
4	3	biimpd 232	. . . . . . . . 9 ⊢ (𝜑 → (if-(𝜑, 𝑧 = 𝑎, 𝑧 = 𝑏) → 𝑧 = 𝑎))
5		equtrr 2055	. . . . . . . . 9 ⊢ (𝑎 = 𝑦 → (𝑧 = 𝑎 → 𝑧 = 𝑦))
6	4, 5	sylan9r 518	. . . . . . . 8 ⊢ ((𝑎 = 𝑦 ∧ 𝜑) → (if-(𝜑, 𝑧 = 𝑎, 𝑧 = 𝑏) → 𝑧 = 𝑦))
7	6	alrimiv 1960	. . . . . . 7 ⊢ ((𝑎 = 𝑦 ∧ 𝜑) → ∀𝑧(if-(𝜑, 𝑧 = 𝑎, 𝑧 = 𝑏) → 𝑧 = 𝑦))
8	7	expcom 419	. . . . . 6 ⊢ (𝜑 → (𝑎 = 𝑦 → ∀𝑧(if-(𝜑, 𝑧 = 𝑎, 𝑧 = 𝑏) → 𝑧 = 𝑦)))
9	8	eximdv 1950	. . . . 5 ⊢ (𝜑 → (∃𝑦 𝑎 = 𝑦 → ∃𝑦∀𝑧(if-(𝜑, 𝑧 = 𝑎, 𝑧 = 𝑏) → 𝑧 = 𝑦)))
10	2, 9	mpi 21	. . . 4 ⊢ (𝜑 → ∃𝑦∀𝑧(if-(𝜑, 𝑧 = 𝑎, 𝑧 = 𝑏) → 𝑧 = 𝑦))
11		ax6evr 2048	. . . . 5 ⊢ ∃𝑦 𝑏 = 𝑦
12		ifpfal 1092	. . . . . . . . . 10 ⊢ (¬ 𝜑 → (if-(𝜑, 𝑧 = 𝑎, 𝑧 = 𝑏) ↔ 𝑧 = 𝑏))
13	12	biimpd 232	. . . . . . . . 9 ⊢ (¬ 𝜑 → (if-(𝜑, 𝑧 = 𝑎, 𝑧 = 𝑏) → 𝑧 = 𝑏))
14		equtrr 2055	. . . . . . . . 9 ⊢ (𝑏 = 𝑦 → (𝑧 = 𝑏 → 𝑧 = 𝑦))
15	13, 14	sylan9r 518	. . . . . . . 8 ⊢ ((𝑏 = 𝑦 ∧ ¬ 𝜑) → (if-(𝜑, 𝑧 = 𝑎, 𝑧 = 𝑏) → 𝑧 = 𝑦))
16	15	alrimiv 1960	. . . . . . 7 ⊢ ((𝑏 = 𝑦 ∧ ¬ 𝜑) → ∀𝑧(if-(𝜑, 𝑧 = 𝑎, 𝑧 = 𝑏) → 𝑧 = 𝑦))
17	16	expcom 419	. . . . . 6 ⊢ (¬ 𝜑 → (𝑏 = 𝑦 → ∀𝑧(if-(𝜑, 𝑧 = 𝑎, 𝑧 = 𝑏) → 𝑧 = 𝑦)))
18	17	eximdv 1950	. . . . 5 ⊢ (¬ 𝜑 → (∃𝑦 𝑏 = 𝑦 → ∃𝑦∀𝑧(if-(𝜑, 𝑧 = 𝑎, 𝑧 = 𝑏) → 𝑧 = 𝑦)))
19	11, 18	mpi 21	. . . 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		dfmo 2570	. . 3 ⊢ (∃*𝑧if-(𝜑, 𝑧 = 𝑎, 𝑧 = 𝑏) ↔ ∃𝑦∀𝑧(if-(𝜑, 𝑧 = 𝑎, 𝑧 = 𝑏) → 𝑧 = 𝑦))
22	20, 21	mpbir 234	. 2 ⊢ ∃*𝑧if-(𝜑, 𝑧 = 𝑎, 𝑧 = 𝑏)
23	1, 22	mpg 1830	1 ⊢ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑥 if-(𝜑, 𝑧 = 𝑎, 𝑧 = 𝑏))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401  if-wif 1078  ∀wal 1568  ∃wex 1812  ∃*wmo 2567  ∃wrex 3091

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-rep 5240

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ifp 1079  df-ex 1813  df-mo 2569  df-rex 3092

This theorem is used by: (None)