Theorem seqeq2d 13972

Description: Equality deduction for the sequence builder operation. (Contributed by Mario Carneiro, 7-Sep-2013.)

Hypothesis

Ref	Expression
seqeqd.1	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

Ref	Expression
seqeq2d	⊢ (𝜑 → seq𝑀(𝐴, 𝐹) = seq𝑀(𝐵, 𝐹))

Proof of Theorem seqeq2d

Step	Hyp	Ref	Expression
1		seqeqd.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		seqeq2 13969	. 2 ⊢ (𝐴 = 𝐵 → seq𝑀(𝐴, 𝐹) = seq𝑀(𝐵, 𝐹))
3	1, 2	syl 17	1 ⊢ (𝜑 → seq𝑀(𝐴, 𝐹) = seq𝑀(𝐵, 𝐹))

Syntax hints:   → wi 4   = wceq 1541  seqcseq 13965

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-xp 5682  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-iota 6495  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-seq 13966

This theorem is referenced by:  seqeq123d  13974  sadfval  16392  smufval  16417  gsumvalx  18594  gsumpropd  18596  gsumress  18600  mulgfval  18951  mulgfvalALT  18952  submmulg  18997  subgmulg  19019  dvnfval  25438