Theorem cbvopab1v 4636

Description: Rule used to change the first bound variable in an ordered pair abstraction, using implicit substitution. (Contributed by NM, 31-Jul-2003.) (Proof shortened by Eric Schmidt, 4-Apr-2007.)

Hypothesis

Ref	Expression
cbvopab1v.1	⊢ (x = z → (φ ↔ ψ))

Assertion

Ref	Expression
cbvopab1v	⊢ {⟨x, y⟩ ∣ φ} = {⟨z, y⟩ ∣ ψ}

Distinct variable groups:   x,y   y,z   φ,z   ψ,x

Allowed substitution hints:   φ(x,y)   ψ(y,z)

Proof of Theorem cbvopab1v

Step	Hyp	Ref	Expression
1		nfv 1619	. 2 ⊢ Ⅎzφ
2		nfv 1619	. 2 ⊢ Ⅎxψ
3		cbvopab1v.1	. 2 ⊢ (x = z → (φ ↔ ψ))
4	1, 2, 3	cbvopab1 4633	1 ⊢ {⟨x, y⟩ ∣ φ} = {⟨z, y⟩ ∣ ψ}

Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642  {copab 4623

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-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-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  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-addc 4379  df-nnc 4380  df-phi 4566  df-op 4567  df-opab 4624

This theorem is referenced by: (None)