Theorem ixpeq2 4355

Description: Equality theorem for infinite Cartesian product.

Assertion

Ref	Expression
ixpeq2	|- (A.x e. A B = C -> X_x e. A B = X_x e. A C)

Proof of Theorem ixpeq2

Step	Hyp	Ref	Expression
1		ss2ixp 4354	. . 3 |- (A.x e. A B (_ C -> X_x e. A B (_ X_x e. A C)
2		ss2ixp 4354	. . 3 |- (A.x e. A C (_ B -> X_x e. A C (_ X_x e. A B)
3	1, 2	anim12i 333	. 2 |- ((A.x e. A B (_ C /\ A.x e. A C (_ B) -> (X_x e. A B (_ X_x e. A C /\ X_x e. A C (_ X_x e. A B))
4		eqss 2077	. . . 4 |- (B = C <-> (B (_ C /\ C (_ B))
5	4	ralbii 1667	. . 3 |- (A.x e. A B = C <-> A.x e. A (B (_ C /\ C (_ B))
6		r19.26 1750	. . 3 |- (A.x e. A (B (_ C /\ C (_ B) <-> (A.x e. A B (_ C /\ A.x e. A C (_ B))
7	5, 6	bitr 173	. 2 |- (A.x e. A B = C <-> (A.x e. A B (_ C /\ A.x e. A C (_ B))
8		eqss 2077	. 2 |- (X_x e. A B = X_x e. A C <-> (X_x e. A B (_ X_x e. A C /\ X_x e. A C (_ X_x e. A B))
9	3, 7, 8	3imtr4 219	1 |- (A.x e. A B = C -> X_x e. A B = X_x e. A C)

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956  A.wral 1645   (_ wss 2047  X_cixp 4347

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-fv 3198  df-ixp 4348

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