Theorem symd 5924

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

Hypotheses

Ref	Expression
symd.1	⊢ (φ → R Sym A)
symd.2	⊢ (φ → X ∈ A)
symd.3	⊢ (φ → Y ∈ A)
symd.4	⊢ (φ → XRY)

Assertion

Ref	Expression
symd	⊢ (φ → YRX)

Proof of Theorem symd

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

Step	Hyp	Ref	Expression
1		symd.2	. . 3 ⊢ (φ → X ∈ A)
2		symd.3	. . 3 ⊢ (φ → Y ∈ A)
3	1, 2	jca 518	. 2 ⊢ (φ → (X ∈ A ∧ Y ∈ A))
4		symd.1	. . 3 ⊢ (φ → R Sym A)
5		brex 4689	. . . . 5 ⊢ (R Sym A → (R ∈ V ∧ A ∈ V))
6		breq 4641	. . . . . . . 8 ⊢ (r = R → (xry ↔ xRy))
7		breq 4641	. . . . . . . 8 ⊢ (r = R → (yrx ↔ yRx))
8	6, 7	imbi12d 311	. . . . . . 7 ⊢ (r = R → ((xry → yrx) ↔ (xRy → yRx)))
9	8	2ralbidv 2656	. . . . . 6 ⊢ (r = R → (∀x ∈ a ∀y ∈ a (xry → yrx) ↔ ∀x ∈ a ∀y ∈ a (xRy → yRx)))
10		raleq 2807	. . . . . . 7 ⊢ (a = A → (∀y ∈ a (xRy → yRx) ↔ ∀y ∈ A (xRy → yRx)))
11	10	raleqbi1dv 2815	. . . . . 6 ⊢ (a = A → (∀x ∈ a ∀y ∈ a (xRy → yRx) ↔ ∀x ∈ A ∀y ∈ A (xRy → yRx)))
12		df-sym 5908	. . . . . 6 ⊢ Sym = {⟨r, a⟩ ∣ ∀x ∈ a ∀y ∈ a (xry → yrx)}
13	9, 11, 12	brabg 4706	. . . . 5 ⊢ ((R ∈ V ∧ A ∈ V) → (R Sym A ↔ ∀x ∈ A ∀y ∈ A (xRy → yRx)))
14	5, 13	syl 15	. . . 4 ⊢ (R Sym A → (R Sym A ↔ ∀x ∈ A ∀y ∈ A (xRy → yRx)))
15	14	ibi 232	. . 3 ⊢ (R Sym A → ∀x ∈ A ∀y ∈ A (xRy → yRx))
16	4, 15	syl 15	. 2 ⊢ (φ → ∀x ∈ A ∀y ∈ A (xRy → yRx))
17		symd.4	. 2 ⊢ (φ → XRY)
18		breq1 4642	. . . 4 ⊢ (x = X → (xRy ↔ XRy))
19		breq2 4643	. . . 4 ⊢ (x = X → (yRx ↔ yRX))
20	18, 19	imbi12d 311	. . 3 ⊢ (x = X → ((xRy → yRx) ↔ (XRy → yRX)))
21		breq2 4643	. . . 4 ⊢ (y = Y → (XRy ↔ XRY))
22		breq1 4642	. . . 4 ⊢ (y = Y → (yRX ↔ YRX))
23	21, 22	imbi12d 311	. . 3 ⊢ (y = Y → ((XRy → yRX) ↔ (XRY → YRX)))
24	20, 23	rspc2v 2961	. 2 ⊢ ((X ∈ A ∧ Y ∈ A) → (∀x ∈ A ∀y ∈ A (xRy → yRx) → (XRY → YRX)))
25	3, 16, 17, 24	syl3c 57	1 ⊢ (φ → YRX)

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∀wral 2614  Vcvv 2859   class class class wbr 4639   Sym csym 5897

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-sym 5908

This theorem is referenced by:  ersym  5952