Theorem prodeq1i 14648

Description: Equality inference for product. (Contributed by Scott Fenton, 4-Dec-2017.)

Hypothesis

Ref	Expression
prodeq1i.1	|- A = B

Assertion

Ref	Expression
prodeq1i	|- prod_ k e. A C = prod_ k e. B C

Distinct variable groups:   A, k   B, k

Allowed substitution hint:   C( k)

Proof of Theorem prodeq1i

Step	Hyp	Ref	Expression
1		prodeq1i.1	. 2 |- A = B
2		prodeq1 14639	. 2 |- ( A = B -> prod_ k e. A C = prod_ k e. B C )
3	1, 2	ax-mp 5	1 |- prod_ k e. A C = prod_ k e. B C

Syntax hints:   = wceq 1483   prod_cprod 14635

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-iota 5851  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-seq 12802  df-prod 14636

This theorem is referenced by:  prodeq12i  14650  fprodxp  14712  risefac0  14758  fallfacfwd  14767  prmo0  15740  breprexp  30711  etransclem31  40482  etransclem35  40486  hoidmv1le  40808  fmtnorec2  41455