Theorem eqerlem 5961

Description: Lemma for eqer 5962. (Contributed by set.mm contributors, 17-Mar-2008.)

Hypotheses

Ref	Expression
eqer.1	⊢ (x = y → A = B)
eqer.2	⊢ R = {⟨x, y⟩ ∣ A = B}

Assertion

Ref	Expression
eqerlem	⊢ (zRw ↔ [z / x]A = [w / x]A)

Distinct variable groups:   y,A   x,B   z,w,R   x,w,y,z

Allowed substitution hints:   A(x,z,w)   B(y,z,w)   R(x,y)

Proof of Theorem eqerlem

Step	Hyp	Ref	Expression
1		eqer.2	. . . 4 ⊢ R = {⟨x, y⟩ ∣ A = B}
2	1	brabsb 4699	. . 3 ⊢ (zRw ↔ [̣z / x]̣[̣w / y]̣A = B)
3		sbccom 3118	. . 3 ⊢ ([̣z / x]̣[̣w / y]̣A = B ↔ [̣w / y]̣[̣z / x]̣A = B)
4	2, 3	bitri 240	. 2 ⊢ (zRw ↔ [̣w / y]̣[̣z / x]̣A = B)
5		vex 2863	. . . . 5 ⊢ z ∈ V
6		sbceq1g 3157	. . . . 5 ⊢ (z ∈ V → ([̣z / x]̣A = B ↔ [z / x]A = B))
7	5, 6	ax-mp 5	. . . 4 ⊢ ([̣z / x]̣A = B ↔ [z / x]A = B)
8		vex 2863	. . . . . 6 ⊢ y ∈ V
9		nfcv 2490	. . . . . 6 ⊢ ℲxB
10		eqer.1	. . . . . 6 ⊢ (x = y → A = B)
11	8, 9, 10	csbief 3178	. . . . 5 ⊢ [y / x]A = B
12	11	eqeq2i 2363	. . . 4 ⊢ ([z / x]A = [y / x]A ↔ [z / x]A = B)
13	7, 12	bitr4i 243	. . 3 ⊢ ([̣z / x]̣A = B ↔ [z / x]A = [y / x]A)
14	13	sbcbii 3102	. 2 ⊢ ([̣w / y]̣[̣z / x]̣A = B ↔ [̣w / y]̣[z / x]A = [y / x]A)
15		vex 2863	. . . 4 ⊢ w ∈ V
16		sbceq2g 3159	. . . 4 ⊢ (w ∈ V → ([̣w / y]̣[z / x]A = [y / x]A ↔ [z / x]A = [w / y][y / x]A))
17	15, 16	ax-mp 5	. . 3 ⊢ ([̣w / y]̣[z / x]A = [y / x]A ↔ [z / x]A = [w / y][y / x]A)
18		csbco 3146	. . . 4 ⊢ [w / y][y / x]A = [w / x]A
19	18	eqeq2i 2363	. . 3 ⊢ ([z / x]A = [w / y][y / x]A ↔ [z / x]A = [w / x]A)
20	17, 19	bitri 240	. 2 ⊢ ([̣w / y]̣[z / x]A = [y / x]A ↔ [z / x]A = [w / x]A)
21	4, 14, 20	3bitri 262	1 ⊢ (zRw ↔ [z / x]A = [w / x]A)

Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642   ∈ wcel 1710  Vcvv 2860  [̣wsbc 3047  [csb 3137  {copab 4623   class class class wbr 4640

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 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-csb 3138  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641

This theorem is referenced by:  eqer  5962