Theorem extd 5923

Description: Extensional relationship in natural deduction form. (Contributed by SF, 20-Feb-2015.)

Hypotheses

Ref	Expression
extd.1	⊢ (φ → R Ext A)
extd.2	⊢ (φ → X ∈ A)
extd.3	⊢ (φ → Y ∈ A)
extd.4	⊢ ((φ ∧ z ∈ A) → (zRX ↔ zRY))

Assertion

Ref	Expression
extd	⊢ (φ → X = Y)

Distinct variable groups:   z,A   φ,z   z,R   z,X   z,Y

Proof of Theorem extd

Dummy variables a r x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		extd.2	. . 3 ⊢ (φ → X ∈ A)
2		extd.3	. . 3 ⊢ (φ → Y ∈ A)
3	1, 2	jca 518	. 2 ⊢ (φ → (X ∈ A ∧ Y ∈ A))
4		extd.1	. . 3 ⊢ (φ → R Ext A)
5		brex 4689	. . . . 5 ⊢ (R Ext A → (R ∈ V ∧ A ∈ V))
6		breq 4641	. . . . . . . . . 10 ⊢ (r = R → (zrx ↔ zRx))
7		breq 4641	. . . . . . . . . 10 ⊢ (r = R → (zry ↔ zRy))
8	6, 7	bibi12d 312	. . . . . . . . 9 ⊢ (r = R → ((zrx ↔ zry) ↔ (zRx ↔ zRy)))
9	8	ralbidv 2634	. . . . . . . 8 ⊢ (r = R → (∀z ∈ a (zrx ↔ zry) ↔ ∀z ∈ a (zRx ↔ zRy)))
10	9	imbi1d 308	. . . . . . 7 ⊢ (r = R → ((∀z ∈ a (zrx ↔ zry) → x = y) ↔ (∀z ∈ a (zRx ↔ zRy) → x = y)))
11	10	2ralbidv 2656	. . . . . 6 ⊢ (r = R → (∀x ∈ a ∀y ∈ a (∀z ∈ a (zrx ↔ zry) → x = y) ↔ ∀x ∈ a ∀y ∈ a (∀z ∈ a (zRx ↔ zRy) → x = y)))
12		raleq 2807	. . . . . . . . 9 ⊢ (a = A → (∀z ∈ a (zRx ↔ zRy) ↔ ∀z ∈ A (zRx ↔ zRy)))
13	12	imbi1d 308	. . . . . . . 8 ⊢ (a = A → ((∀z ∈ a (zRx ↔ zRy) → x = y) ↔ (∀z ∈ A (zRx ↔ zRy) → x = y)))
14	13	raleqbi1dv 2815	. . . . . . 7 ⊢ (a = A → (∀y ∈ a (∀z ∈ a (zRx ↔ zRy) → x = y) ↔ ∀y ∈ A (∀z ∈ A (zRx ↔ zRy) → x = y)))
15	14	raleqbi1dv 2815	. . . . . 6 ⊢ (a = A → (∀x ∈ a ∀y ∈ a (∀z ∈ a (zRx ↔ zRy) → x = y) ↔ ∀x ∈ A ∀y ∈ A (∀z ∈ A (zRx ↔ zRy) → x = y)))
16		df-ext 5907	. . . . . 6 ⊢ Ext = {⟨r, a⟩ ∣ ∀x ∈ a ∀y ∈ a (∀z ∈ a (zrx ↔ zry) → x = y)}
17	11, 15, 16	brabg 4706	. . . . 5 ⊢ ((R ∈ V ∧ A ∈ V) → (R Ext A ↔ ∀x ∈ A ∀y ∈ A (∀z ∈ A (zRx ↔ zRy) → x = y)))
18	5, 17	syl 15	. . . 4 ⊢ (R Ext A → (R Ext A ↔ ∀x ∈ A ∀y ∈ A (∀z ∈ A (zRx ↔ zRy) → x = y)))
19	18	ibi 232	. . 3 ⊢ (R Ext A → ∀x ∈ A ∀y ∈ A (∀z ∈ A (zRx ↔ zRy) → x = y))
20	4, 19	syl 15	. 2 ⊢ (φ → ∀x ∈ A ∀y ∈ A (∀z ∈ A (zRx ↔ zRy) → x = y))
21		extd.4	. . 3 ⊢ ((φ ∧ z ∈ A) → (zRX ↔ zRY))
22	21	ralrimiva 2697	. 2 ⊢ (φ → ∀z ∈ A (zRX ↔ zRY))
23		breq2 4643	. . . . . 6 ⊢ (x = X → (zRx ↔ zRX))
24	23	bibi1d 310	. . . . 5 ⊢ (x = X → ((zRx ↔ zRy) ↔ (zRX ↔ zRy)))
25	24	ralbidv 2634	. . . 4 ⊢ (x = X → (∀z ∈ A (zRx ↔ zRy) ↔ ∀z ∈ A (zRX ↔ zRy)))
26		eqeq1 2359	. . . 4 ⊢ (x = X → (x = y ↔ X = y))
27	25, 26	imbi12d 311	. . 3 ⊢ (x = X → ((∀z ∈ A (zRx ↔ zRy) → x = y) ↔ (∀z ∈ A (zRX ↔ zRy) → X = y)))
28		breq2 4643	. . . . . 6 ⊢ (y = Y → (zRy ↔ zRY))
29	28	bibi2d 309	. . . . 5 ⊢ (y = Y → ((zRX ↔ zRy) ↔ (zRX ↔ zRY)))
30	29	ralbidv 2634	. . . 4 ⊢ (y = Y → (∀z ∈ A (zRX ↔ zRy) ↔ ∀z ∈ A (zRX ↔ zRY)))
31		eqeq2 2362	. . . 4 ⊢ (y = Y → (X = y ↔ X = Y))
32	30, 31	imbi12d 311	. . 3 ⊢ (y = Y → ((∀z ∈ A (zRX ↔ zRy) → X = y) ↔ (∀z ∈ A (zRX ↔ zRY) → X = Y)))
33	27, 32	rspc2v 2961	. 2 ⊢ ((X ∈ A ∧ Y ∈ A) → (∀x ∈ A ∀y ∈ A (∀z ∈ A (zRx ↔ zRy) → x = y) → (∀z ∈ A (zRX ↔ zRY) → X = Y)))
34	3, 20, 22, 33	syl3c 57	1 ⊢ (φ → X = Y)

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∀wral 2614  Vcvv 2859   class class class wbr 4639   Ext cext 5896

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-ext 5907

This theorem is referenced by: (None)