Theorem cbvmpt2 5679

Description: Rule to change the bound variable in a maps-to function, using implicit substitution. (Contributed by NM, 17-Dec-2013.)

Hypotheses

Ref	Expression
cbvmpt2.1	⊢ ℲzC
cbvmpt2.2	⊢ ℲwC
cbvmpt2.3	⊢ ℲxD
cbvmpt2.4	⊢ ℲyD
cbvmpt2.5	⊢ ((x = z ∧ y = w) → C = D)

Assertion

Ref	Expression
cbvmpt2	⊢ (x ∈ A, y ∈ B ↦ C) = (z ∈ A, w ∈ B ↦ D)

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

Allowed substitution hints:   C(x,y,z,w)   D(x,y,z,w)

Proof of Theorem cbvmpt2

Step	Hyp	Ref	Expression
1		nfcv 2489	. 2 ⊢ ℲzB
2		nfcv 2489	. 2 ⊢ ℲxB
3		cbvmpt2.1	. 2 ⊢ ℲzC
4		cbvmpt2.2	. 2 ⊢ ℲwC
5		cbvmpt2.3	. 2 ⊢ ℲxD
6		cbvmpt2.4	. 2 ⊢ ℲyD
7		eqidd 2354	. 2 ⊢ (x = z → B = B)
8		cbvmpt2.5	. 2 ⊢ ((x = z ∧ y = w) → C = D)
9	1, 2, 3, 4, 5, 6, 7, 8	cbvmpt2x 5678	1 ⊢ (x ∈ A, y ∈ B ↦ C) = (z ∈ A, w ∈ B ↦ D)

Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642  Ⅎwnfc 2476   ↦ cmpt2 5653

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 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-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 2478  df-ne 2518  df-ral 2619  df-rex 2620  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-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-addc 4378  df-nnc 4379  df-phi 4565  df-op 4566  df-opab 4623  df-oprab 5528  df-mpt2 5654

This theorem is referenced by:  cbvmpt2v  5680